explain that X-rays are produced by electron bombardment of a metal target and calculate the minimum wavelength of X-rays produced from the accelerating p.d.

Production and Use of X‑rays – Cambridge International AS & A Level Physics (9702)

Learning Objectives

• Explain how high‑energy electrons produce X‑rays when they strike a metal target.
• Describe the two radiative processes: bremsstrahlung and characteristic radiation.
• Derive and apply the relationship \(\displaystyle \lambda_{\min}= \frac{hc}{eV}\) to find the shortest wavelength for a given accelerating potential.
• Use the attenuation law \(\displaystyle I = I_{0}e^{-\mu x}\) and the concept of half‑value layer (HVL) to predict intensity loss in different materials.
• Summarise the principle of CT scanning and the factors that affect image contrast.
• Identify the main safety, shielding and dose‑management requirements (ALARA) when working with X‑ray equipment.

1. How X‑rays are Generated

1.1 Electron emission and acceleration

• Thermionic emission: a heated cathode (filament) releases electrons.
• Accelerating potential \(V\): a high voltage (typically 10–150 kV) creates an electric field that accelerates the electrons toward the anode.

  
  Each electron gains kinetic energy \(\displaystyle K = eV\) (where \(e = 1.602\times10^{-19}\,\text{C}\)).

1.2 Interaction with the metal target (anode)

When the fast electrons strike the target nuclei two radiative processes occur:

• Bremsstrahlung (braking radiation)

  • Electrons are decelerated in the strong electric field of the nuclei.
  • The loss of kinetic energy is emitted as photons, giving a continuous spectrum from \(\lambda_{\min}\) up to several nanometres.
  • The intensity of bremsstrahlung increases with the atomic number \(Z\) of the target (higher \(Z\) → stronger electric field).

• Characteristic radiation

  • Incident electrons may eject an inner‑shell (K or L) electron from the target atom.
  • Electrons from higher shells fall into the vacancy, releasing photons with discrete energies:

    \[

    E{K\alpha}=E{K}-E_{L},\qquad

    E{K\beta}=E{K}-E_{M},\; \text{etc.}

  • The photon energies (and thus wavelengths) depend only on the target’s atomic number \(Z\); they appear as sharp lines superimposed on the bremsstrahlung background.
  • Common targets: Mo (K\(\alpha\)=17.5 keV), Cu (K\(\alpha\)=8.0 keV), W (K\(_\alpha\)=59 keV).

1.3 Tube design considerations

• Target material: high‑\(Z\) metals (W, Mo, Cu) give stronger bremsstrahlung and more intense characteristic lines.
• Heat load: most kinetic energy is converted to heat; the anode is often angled and water‑cooled to spread the heat.
• Efficiency: only ~1 % of the electron energy emerges as X‑rays; the rest is heat.

2. Minimum (Shortest) Wavelength of the Produced X‑rays

Using \(h = 6.626\times10^{-34}\,\text{J·s}\), \(c = 3.00\times10^{8}\,\text{m·s}^{-1}\) and \(e = 1.602\times10^{-19}\,\text{C}\):

\[

\lambda_{\min}= \frac{1240\ \text{nm·kV}}{V\ (\text{kV})}

\]

Worked Example 1 – 30 kV tube

\[

\lambda_{\min}= \frac{1240}{30}\ \text{nm}=41.3\ \text{pm}=0.041\ \text{nm}

\]

Worked Example 2 – 80 kV tube (medical CT)

\[

\lambda_{\min}= \frac{1240}{80}\ \text{nm}=15.5\ \text{pm}=0.0155\ \text{nm}

\]

3. Attenuation of X‑rays in Matter

3.1 Exponential attenuation

\[

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

\]

• \(I_{0}\) – incident intensity.
• \(I\) – transmitted intensity after travelling a distance \(x\) (cm) in the material.
• \(\mu\) – linear attenuation coefficient (cm\(^{-1}\)), which depends on photon energy and the atomic composition of the material.

3.2 Half‑value layer (HVL)

The thickness that reduces the intensity to one half is

\[

\text{HVL}= \frac{\ln 2}{\mu}

\]

3.3 Worked Example – Aluminium filter (30 keV)

• \(\mu_{\text{Al}} = 0.55\ \text{cm}^{-1}\)
• Thickness \(x = 2\ \text{mm}=0.20\ \text{cm}\)

\[

\frac{I}{I_{0}} = e^{-\mu x}=e^{-0.55\times0.20}=e^{-0.11}=0.895

\]

≈ 90 % transmitted, 10 % absorbed.

3.4 Worked Example – Lead filter (80 keV)

• \(\mu_{\text{Pb}} = 4.0\ \text{cm}^{-1}\) (approx. for 80 keV photons)
• Thickness \(x = 0.1\ \text{cm}=1\ \text{mm}\)

\[

\frac{I}{I_{0}} = e^{-4.0\times0.10}=e^{-0.40}=0.670

\]

≈ 67 % transmitted.

HVL for the same case

\[

\text{HVL}= \frac{\ln 2}{4.0}=0.173\ \text{cm}=1.73\ \text{mm}

\]

4. Computed Tomography (CT) Scanning

• Principle: an X‑ray tube and detector rotate around the patient, acquiring many projection images at different angles. Reconstruction algorithms (filtered back‑projection or iterative methods) combine the data to produce cross‑sectional images.
• Image contrast: depends on differences in linear attenuation coefficients \(\mu\) of tissues.

  • Low‑\(V\) (e.g., 80 kV) → larger \(\mu\) differences → higher contrast but reduced penetration.
  • Contrast agents (iodine, barium) raise the local \(\mu\) and enhance visibility of vessels or gastrointestinal tracts.
  • Reconstruction parameters (slice thickness, filter) also affect perceived contrast.

5. Safety, Shielding and Dose Management (ALARA)

• Time, distance, shielding – minimise exposure time, maximise distance, use appropriate shielding.
• Shielding materials

  • Lead (Pb): most common; thickness chosen from \(\mu\) values (e.g., 0.5 mm Pb reduces 100 keV X‑rays by ≈ 90 %).
  • Concrete or steel: used for walls in radiology rooms.

• Personal protective equipment – lead aprons, thyroid collars, lead glasses.
• Regulatory dose limits (ICRP)

  • Occupational: 20 mSv yr\(^{-1}\) (average), 50 mSv yr\(^{-1}\) for pregnant workers.
  • Public: 1 mSv yr\(^{-1}\) (excluding medical exposure).

• ALARA principle – keep radiation “As Low As Reasonably Achievable” by optimisation of tube voltage, exposure time, collimation and use of automatic exposure control.
• Engineering controls – interlock doors, warning lights, emergency shut‑off, and regular tube‑output testing.

6. Typical Minimum Wavelengths for Common Tube Voltages

7. Summary

• High‑energy electrons, accelerated through a potential \(V\), strike a metal target and produce X‑rays via bremsstrahlung (continuous spectrum) and characteristic radiation (discrete lines).
• The shortest possible wavelength is \(\displaystyle \lambda{\min}=hc/eV\); the convenient form \(\lambda{\min}(\text{nm})=1240/V(\text{kV})\) allows rapid calculation.
• Intensity attenuation follows \(I=I_{0}e^{-\mu x}\); the half‑value layer \(\text{HVL}= \ln 2/\mu\) provides a handy measure of shielding effectiveness.
• CT scanners acquire many angular projections; image contrast is governed by differences in \(\mu\) and can be enhanced with lower tube voltage or contrast agents.
• Because X‑rays are ionising, strict safety protocols—time‑distance‑shielding, lead protection, interlocks, dose limits, and the ALARA principle—must be observed.

8. Suggested Diagram