recall that a tracer that decays by β+ decay is used in positron emission tomography (PET scanning)
Production and Use of X‑rays (Cambridge IGCSE/A‑Level 9702 – Section 24.2)
1. Production mechanisms
- Bremsstrahlung (braking radiation) – Fast electrons are decelerated in the electric field of nuclei in the target (anode). Their kinetic energy is converted into a continuous spectrum of X‑ray photons ranging from 0 up to a maximum energy.
- Characteristic radiation – An incident electron ejects an inner‑shell electron (usually from the K‑shell). An electron from a higher shell (L, M…) drops down to fill the vacancy and a photon is emitted whose energy equals the difference between the two atomic energy levels.
- Typical lines: K‑α (transition L→K) and K‑β (transition M→K).
- Example: for a copper (Cu) target, K‑α ≈ 8.0 keV and K‑β ≈ 8.9 keV.
Typical proportion in a clinical X‑ray tube: about 90 % Bremsstrahlung photons and 10 % characteristic photons, the exact ratio depending on the target material and tube voltage.
Simple schematic of an X‑ray tube (cathode → filament, anode → target, high‑voltage supply):
2. Link to the accelerating potential
The maximum photon energy (and therefore the minimum wavelength) is limited by the kinetic energy gained by the electrons in the tube voltage \(V\):
\[
E{\max}=eV\qquad\Longrightarrow\qquad \lambda{\min }=\frac{hc}{eV}
\]
Derivation of the convenient numerical form:
\[
\lambda_{\min }(\text{nm})=\frac{hc}{eV}
=\frac{(6.626\times10^{-34}\,\text{J·s})(3.00\times10^{8}\,\text{m s}^{-1})}
{(1.602\times10^{-19}\,\text{C})(V\;\text{V})}
=\frac{1240}{V(\text{kV})}\;\text{nm}
\]
Example – tube voltage 100 kV:
\[
\lambda_{\min }=\frac{1240}{100}=0.0124\;\text{nm}
\]
3. Attenuation of X‑rays in matter
The intensity of a mono‑energetic X‑ray beam after passing through a material of thickness \(x\) is described by the exponential attenuation law:
\[
I = I_{0}\,e^{-\mu x}
\]
- \(I_{0}\) – incident intensity
- \(I\) – transmitted intensity
- \(\mu\) – linear attenuation coefficient (units cm\(^{-1}\) or m\(^{-1}\))
- \(x\) – path length through the material
Factors influencing \(\mu\)
- Atomic number \(Z\) – higher \(Z\) gives larger photoelectric absorption.
- Density \(\rho\) – more atoms per unit volume increase attenuation.
- Photon energy – \(\mu\) falls sharply with increasing energy above the K‑edge of the material.
Quantitative example
| Material | \(\mu\) for 100 keV X‑rays (cm\(^{-1}\)) | Thickness \(x\) (cm) | Transmitted fraction \(I/I_{0}\) |
|---|---|---|---|
| Soft tissue (≈ water) | 0.17 | 10 | \(e^{-0.17\times10}=e^{-1.7}=0.18\) (18 %) |
| Bone (cortical) | 0.55 | 10 | \(e^{-0.55\times10}=e^{-5.5}=0.004\) (0.4 %) |
| Lead (Pb) | 5.0 | 0.2 | \(e^{-5.0\times0.2}=e^{-1}=0.37\) (37 %) |
Thus high‑\(Z\) materials such as bone or lead attenuate X‑rays much more strongly than low‑\(Z\) soft tissue, creating the contrast seen on radiographs.
4. Imaging applications
4.1 Medical radiography
- Single‑direction X‑ray beam passes through the patient.
- Differences in \(\mu\) between tissues produce varying transmitted intensities that are recorded on film or a digital detector.
- Typical clinical tube voltages: 60–120 kV for chest, 70–90 kV for extremities.
4.2 Computed Tomography (CT)
- Series of X‑ray projections are acquired while the X‑ray tube and detector rotate around the patient.
- Reconstruction algorithms (filtered back‑projection or iterative methods) combine the projections to give cross‑sectional images.
- CT numbers (Hounsfield units) are defined as \(\displaystyle \text{HU}=1000\frac{\mu{\text{tissue}}-\mu{\text{water}}}{\mu_{\text{water}}}\).
- Provides quantitative density information and superior contrast resolution compared with plain radiography.
4.3 Industrial non‑destructive testing (NDT)
- Radiography of welds, castings, aerospace components, etc.
- Defects such as cracks, voids or inclusions are identified because they locally change the attenuation of the X‑ray beam.
- High‑energy (MeV) X‑rays are often used to penetrate thick metal sections.
5. Therapeutic use of X‑rays
- High‑energy X‑rays (typically 6–25 MeV) are delivered in precisely controlled doses to tumours.
- The ionising radiation creates DNA damage (single‑ and double‑strand breaks) that inhibits cancer cell division.
- Modern techniques:
- 3‑D conformal radiotherapy – shapes the beam to the tumour volume.
- Intensity‑Modulated Radiotherapy (IMRT) – varies beam intensity across the field.
- Stereotactic radiosurgery – delivers a high dose in a single or few fractions with sub‑millimetre accuracy.
6. Safety and shielding
- Shielding materials: Lead (Pb) is most common because of its high \(\mu\); concrete or steel are used for very high‑energy beams.
- Distance: Radiation intensity follows the inverse‑square law, \(I\propto 1/r^{2}\); increasing the distance from the source rapidly reduces exposure.
- ALARA principle: All exposures should be kept “As Low As Reasonably Achievable” by combining shielding, distance, and limiting exposure time.
- Dose limits (ICRP recommendations, widely adopted in the UK):
- Public: 1 mSv yr\(^{-1}\) (excluding natural background).
- Occupational (radiographers, radiotherapists): 20 mSv yr\(^{-1}\) averaged over 5 years, with no single year exceeding 50 mSv.
7. Positron Emission Tomography (PET) – optional extension
Although PET detects 511 keV annihilation photons (γ‑rays), it is taught alongside X‑ray topics because it uses the same scintillation detector technology.
- Radioactive tracer undergoes β⁺ decay:
\[
\ce{{Z}^{A}X -> {Z-1}^{A}Y + e^{+} + \nu_e}
\]
- The emitted positron travels a few millimetres in tissue, then annihilates with an electron, producing two photons of 511 keV emitted almost exactly 180° apart.
- Coincidence detection: a pair of detectors surrounding the patient records the simultaneous arrival of the two photons. By drawing a line of response (LOR) between the detectors, the location of the annihilation event is constrained to that line.
- Repeated LORs are fed into reconstruction algorithms (filtered back‑projection, OSEM) to produce a three‑dimensional map of tracer concentration, revealing metabolic activity.
| Tracer (commercial name) | Radioisotope | Half‑life | Decay mode | Typical clinical use |
|---|---|---|---|---|
| ⁱ⁸F‑FDG | Fluorine‑18 | 110 min | β⁺ | Oncological imaging, brain glucose metabolism |
| ⁸⁸Y‑DOTATATE | Yttrium‑88 | 106 min | β⁺ | Neuroendocrine tumour imaging |
| ¹¹C‑PiB | Carbon‑11 | 20 min | β⁺ | Alzheimer’s disease amyloid imaging |
| ⁶⁸Ga‑DOTATOC | Gallium‑68 | 68 min | β⁺ | Neuroendocrine tumours, somatostatin receptor imaging |
8. Summary of learning objectives
- Identify and describe the two X‑ray production mechanisms (Bremsstrahlung and characteristic radiation) and give typical line energies (e.g., Cu K‑α ≈ 8 keV).
- Derive and use the relation \(\lambda_{\min}=1240/V(\text{kV})\) nm; calculate minimum wavelength for a given tube voltage.
- Apply the attenuation law \(I=I_{0}e^{-\mu x}\); calculate transmitted intensity for given \(\mu\) and thickness, and explain why high‑\(Z\) materials appear bright on radiographs.
- Distinguish the main imaging modalities (plain radiography, CT, industrial NDT) and state the typical operating voltages or energies.
- Explain how high‑energy X‑rays are used therapeutically and name at least two modern radiotherapy techniques.
- Recall the key safety principles: appropriate shielding, distance (inverse‑square law), and the ALARA concept, together with the ICRP dose limits for public and occupational exposure.
- (Optional) State why a β⁺‑decaying tracer is required for PET, describe the annihilation process, and outline the principle of coincidence detection and image reconstruction.