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: \[ \ce{_{Z}^{A}X -> _{Z-1}^{A}Y + e^{+} + u_e} \]
• 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.