understand that a tracer is a substance containing radioactive nuclei that can be introduced into the body and is then absorbed by the tissue being studied

1. Key concepts required by the Cambridge International AS & A Level Physics (9702) syllabus

The material on X‑rays and radioactive tracers links directly to the following syllabus ideas:

• Models and testing – bremsstrahlung and characteristic‑X‑ray models; experimental verification of the attenuation law.
• Mathematics – use of equations for energy, voltage, exponential decay and attenuation, and error propagation.
• Forces and fields – electric fields that accelerate electrons; magnetic fields that steer the electron beam (Lorentz force).
• Quantum and nuclear physics – photon energy–frequency relation, photoelectric effect, Compton scattering, radioactive decay, half‑life and activity.
• Practical skills (Paper 3 & 5) – designing experiments, measuring quantities, analysing uncertainties, evaluating results.

2. X‑rays – production, properties and quantitative description

2.1 What an X‑ray is

X‑rays are high‑energy electromagnetic photons with wavelengths between ≈ 0.01 nm and 10 nm (energies 0.1–100 keV). Their ability to be absorbed differently by various tissues makes them essential for medical imaging.

2.2 Energy–frequency–wavelength relations (quantum physics)

• Photon energy: \(E = h\nu\) where \(h = 6.626\times10^{-34}\ \text{J·s}\).
• Wavelength: \(\displaystyle \lambda = \frac{c}{\nu}= \frac{hc}{E}\).
• Typical values: \(E=30\ \text{keV}\Rightarrow \lambda\approx0.041\ \text{nm}\).

2.3 How X‑rays are generated in an X‑ray tube

1. Electron source (cathode) – a heated filament emits electrons by thermionic emission (link to AS topic 9 & 10: work function, electric current).
2. Accelerating electric field – a high voltage (30–150 kV) creates an electric field \(E = V/d\) that accelerates electrons. The kinetic energy of an electron is

   \[

   E_{\text{k}} = eV

   \]

   where \(e = 1.60\times10^{-19}\ \text{C}\).

3. Magnetic focusing – a solenoidal magnetic field \(\mathbf{B}\) exerts a Lorentz force \(\mathbf{F}=e\mathbf{v}\times\mathbf{B}\) to keep the electron beam centred on the target (connects to A‑level extension topic 15 – magnetic fields).
4. Target (anode) – usually tungsten (high \(Z\), high melting point). Two photon‑production mechanisms occur:

2.3.1 Bremsstrahlung (braking radiation)

When a fast electron is abruptly decelerated in the electric field of a nucleus, its kinetic energy is converted into a continuous spectrum of X‑ray photons. The most energetic photon corresponds to the whole electron energy becoming photon energy, giving the minimum wavelength:

\[

\lambda_{\min}= \frac{hc}{eV}

\]

2.3.2 Characteristic X‑rays

If the incident electron ejects an inner‑shell electron, an outer‑shell electron fills the vacancy and a photon with energy equal to the difference between the two atomic energy levels is emitted. These appear as sharp lines (e.g., Kα, Kβ) superimposed on the bremsstrahlung background.

2.4 Worked example 1 – Minimum wavelength for a typical tube

For an accelerating voltage \(V = 120\ \text{kV}\):

\[

\lambda_{\min}= \frac{6.626\times10^{-34}\times3.00\times10^{8}}

{1.60\times10^{-19}\times120\times10^{3}}

\approx 1.03\times10^{-11}\ \text{m}=0.0103\ \text{nm}

\]

2.5 Alternating‑current (AC) high‑voltage supply

Modern X‑ray units use a step‑up transformer driven by a 50/60 Hz AC source to generate the required kilovolt potential. Understanding AC circuits (impedance, RMS voltage) is part of the A‑level extension on alternating currents.

3. Interaction of X‑rays with matter

3.1 Attenuation law

For a mono‑energetic, narrow beam passing through a material of thickness \(x\):

\[

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

\]

• \(\mu\) – linear attenuation coefficient (units m\(^{-1}\)).
• Often expressed as a mass attenuation coefficient \(\displaystyle \frac{\mu}{\rho}\) (units m\(^2\) kg\(^{-1}\)), which removes the dependence on density \(\rho\).
• \(\mu\) depends strongly on photon energy and the atomic number \(Z\) of the absorber.

3.2 Main interaction processes (10 keV – 1 MeV)

3.3 Quantitative example 2 – Determining \(\mu\) experimentally

Measured transmitted intensities for aluminium:

Using \(I = I_0 e^{-\mu x}\):

\[

\mu = -\frac{1}{x}\ln\!\left(\frac{I}{I_0}\right)

= -\frac{1}{0.005\ \text{m}}\ln\!\left(\frac{45}{120}\right)

\approx 1.33\times10^{2}\ \text{m}^{-1}

\]

Uncertainty can be estimated by propagating the measurement errors on \(I\) and \(I_0\) (Paper 5 skill).

3.4 Quantitative example 3 – Contrast between bone and soft tissue

Assume a 2 cm path through bone (\(\mu{\text{bone}} = 1.5\times10^{2}\ \text{m}^{-1}\)) and the same thickness of muscle (\(\mu{\text{muscle}} = 5.0\times10^{1}\ \text{m}^{-1}\)). With \(I_0 = 1\):

\[

I_{\text{bone}} = e^{-1.5\times10^{2}\times0.02}=e^{-3}=0.050

\]

\[

I_{\text{muscle}} = e^{-5.0\times10^{1}\times0.02}=e^{-1}=0.368

\]

\[

\text{Contrast}= \frac{I{\text{muscle}}-I{\text{bone}}}{I{\text{muscle}}+I{\text{bone}}}

= \frac{0.368-0.050}{0.368+0.050}

\approx 0.76

\]

A high contrast value explains why bone appears white on a radiograph.

4. Medical imaging techniques that use X‑rays

4.1 Radiography (2‑D projection)

• Single exposure; contrast produced by differences in \(\mu\).
• Useful for bone fractures, chest screening.

4.2 Computed Tomography (CT)

• Series of narrow‑beam projections taken while the X‑ray tube rotates.
• Reconstruction (filtered back‑projection) yields cross‑sectional images with high spatial resolution.
• CT numbers (Hounsfield units) are directly related to \(\mu\) of the tissue.

4.3 Fluoroscopy

• Real‑time X‑ray video; low‑dose pulsed mode.
• Guides catheter placement, joint studies, and contrast‑agent studies.

5. Radioactive tracers – nuclear‑physics basis

5.1 Definition (syllabus topic 24 – nuclear physics)

A tracer is a chemically identical compound that contains a short‑lived radioactive nucleus. After administration it is taken up by the organ or tissue of interest, and the emitted radiation is detected to map its distribution.

5.2 Radioactive decay equations

• Activity: \(A = \lambda N\) (\(\lambda\) = decay constant, \(N\) = number of nuclei).
• Half‑life: \(t_{1/2} = \frac{\ln 2}{\lambda}\).
• Number of nuclei after time \(t\): \(N = N_{0}e^{-\lambda t}\).

5.3 Common medical tracers

5.4 Detection methods

• Gamma‑camera (scintillation camera) – NaI(Tl) crystal converts γ‑rays to visible light, which is amplified by a photomultiplier.
• Positron Emission Tomography (PET) – detects the two 511 keV photons emitted back‑to‑back when a positron annihilates with an electron.
• SPECT – rotates a gamma‑camera around the patient to produce tomographic slices.

5.5 Hybrid imaging – PET/CT and SPECT/CT

• CT supplies high‑resolution anatomical information (based on X‑ray attenuation).
• PET or SPECT adds functional data from the tracer (e.g., tumour glucose uptake, myocardial perfusion).
• The combined dataset allows precise localisation of metabolic abnormalities.

6. Safety considerations

6.1 General principles (ALARA)

1. Time – minimise exposure duration.
2. Distance – intensity follows the inverse‑square law \(I\propto 1/r^{2}\).
3. Shielding – lead aprons, concrete walls, beryllium windows for X‑ray tubes, tungsten or lead shielding for radionuclide sources.

6.2 Radiation dose units

• Absorbed dose: gray (Gy) = J kg\(^{-1}\).
• Equivalent dose: sievert (Sv) = Gy × radiation‑weighting factor.
• Effective dose incorporates tissue‑weighting factors to assess risk.

6.3 Tracer‑specific safety

1. Use the lowest activity that gives an acceptable image (dose optimisation).
2. Prefer isotopes with short half‑lives to limit total radiation burden.
3. Follow strict contamination control: lead shields, syringe shields, and proper disposal of radioactive waste.
4. Avoid administration to pregnant patients and very young children unless the clinical benefit outweighs the risk.

7. Practical skills (Paper 3 & 5) – sample activities

7.1 Experiment: Verify the minimum‑wavelength formula

1. Set up an X‑ray tube with a variable high‑voltage supply (30–120 kV).
2. Place a crystal diffraction grating (e.g., NaCl) in the beam path and record the first‑order diffraction angle \(\theta\) for several voltages.
3. Use Bragg’s law \(n\lambda = 2d\sin\theta\) (with known lattice spacing \(d\)) to determine \(\lambda\).
4. Plot \(\lambda\) against \(1/V\); the slope should equal \(hc/e\). Include uncertainties on \(\theta\) and \(V\) and propagate them to the final slope.

7.2 Experiment: Measure linear attenuation coefficients

1. Collimate a mono‑energetic X‑ray beam (e.g., 60 keV) and record the detector current \(I_0\) without any absorber.
2. Insert sheets of aluminium of known thicknesses (1 mm, 2 mm, …) and record the transmitted current \(I\) for each.
3. Plot \(\ln(I/I_0)\) versus thickness \(x\); the gradient gives \(-\mu\).
4. Repeat with a higher‑\(Z\) material (e.g., copper) and discuss the dependence of \(\mu\) on atomic number.
5. Evaluate the experiment: sources of systematic error (beam divergence, detector non‑linearity) and random error (current fluctuations).

7.3 Data analysis skill – Calculating activity from half‑life

Given a 5 MBq sample of \(^{99m}\)Tc at the start of a scan, calculate the activity after 3 h. Use \(A = A0 e^{-\lambda t}\) with \(\lambda = \ln2/t{1/2}\). Include the propagated uncertainty if the half‑life is quoted as \(6.01\pm0.01\) h.

8. Summary

X‑rays are produced when high‑energy electrons are decelerated in a metal target; the resulting bremsstrahlung and characteristic photons obey the quantum relations \(E = h\nu\) and \(\lambda{\min}=hc/eV\). Their interaction with matter—photoelectric absorption, Compton scattering and, at higher energies, pair production—leads to exponential attenuation, described by \(I = I0e^{-\mu x}\). Differences in the attenuation coefficient give the contrast essential for radiography, CT and fluoroscopy.

Radioactive tracers introduce short‑lived nuclei into the body. Their decay (described by half‑life and activity equations) emits γ‑rays or positrons that are detected by gamma‑cameras, PET or SPECT. When combined with CT, hybrid images provide both anatomical detail and functional information, a cornerstone of modern diagnostics.

All techniques are governed by the ALARA principle: minimise time, maximise distance and use appropriate shielding. The experimental methods outlined develop the practical skills required for Cambridge AS & A‑Level Physics, linking theory to measurement, uncertainty analysis and critical evaluation.