Production and use of X-rays

2.1 Overview

X-rays are high-energy electromagnetic waves with wavelengths typically in the range of $10^{-8}$ to $10^{-13}$ m. They are produced when high-speed electrons, accelerated through a significant potential difference, are rapidly decelerated upon impact with a metal target. In medical diagnostics, X-rays are used to create images of internal structures because they can penetrate soft tissue but are absorbed or scattered (attenuated) by denser materials like bone. The fundamental physics of X-ray imaging involves the controlled production of a beam, the exponential attenuation of that beam as it passes through matter, and the detection of the transmitted intensity to form a 2D or 3D representation of the body.

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2.2 Key Definitions

• X-ray Photon: A discrete packet of electromagnetic energy. The energy $E$ is proportional to the frequency $f$ ($E = hf$) and inversely proportional to the wavelength $\lambda$ ($E = \frac{hc}{\lambda}$).
• Attenuation: The gradual reduction in intensity of an X-ray beam as it passes through a medium, caused by the absorption and scattering of photons.
• Linear Attenuation Coefficient ($\mu$): A constant that represents the fraction of X-ray intensity removed per unit thickness of a specific material. It depends on the material's density, atomic number, and the energy of the X-ray photons. (Unit: $\text{m}^{-1}$ or $\text{cm}^{-1}$).
• Half-value Thickness ($x_{1/2}$): The thickness of a material required to reduce the intensity of an incident X-ray beam to exactly half of its original value.
• Contrast: The difference in the degree of blackening (optical density) between different areas of an X-ray image. High contrast means there is a clear, sharp difference between light and dark regions, allowing different tissues to be distinguished.
• Sharpness: The clarity or definition of the edges of the structures in an image. A sharp image has very little "blurring" at the boundaries between different tissues.
• Contrast Medium: A substance (such as Barium or Iodine) with a high atomic number that is introduced into the body to increase the attenuation of X-rays in specific regions (like the digestive tract or blood vessels), thereby improving the contrast of the resulting image.
• Computed Tomography (CT): A diagnostic technique that uses a rotating X-ray source and detectors to take multiple 2D images (slices) from different angles, which are then processed by a computer to reconstruct a 3D image of the internal structure.
• Voxel: A "volume element" representing a value on a regular grid in three-dimensional space, used in the reconstruction of CT images.

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2.3 Content

2.3.1 Production of X-rays

X-rays are generated in an evacuated X-ray tube. The process involves several distinct energy transformations:

1. Thermionic Emission: A low-voltage current heats a tungsten filament (the cathode), causing it to emit electrons.
2. Acceleration: A very high potential difference ($V$), typically between 20 kV and 100 kV, is applied between the cathode and a metal target (the anode). This accelerates the electrons, giving them kinetic energy: $E_k = eV = \frac{1}{2}mv^2$
3. Bombardment: The high-speed electrons strike the metal target (usually Tungsten due to its high melting point).
4. Deceleration and Photon Emission: As electrons hit the target, they interact with the electric fields of the target nuclei and are rapidly decelerated. This loss of kinetic energy results in the emission of X-ray photons.
5. Heat Dissipation: Approximately 99% of the electrons' kinetic energy is converted into thermal energy in the anode. To prevent melting, the anode is often rotated or cooled with oil/water.

The X-ray Spectrum

The output of an X-ray tube consists of a range of wavelengths, forming a characteristic spectrum:

• Continuous Spectrum (Bremsstrahlung): "Braking radiation" produced as electrons decelerate at different rates. An electron might lose all its energy in one collision or only a fraction in several collisions, leading to a continuous range of photon energies.
• Characteristic Peaks: Sharp, high-intensity peaks at specific wavelengths. These occur when an incident electron knocks an inner-shell electron out of a target atom. An electron from a higher energy shell drops down to fill the vacancy, emitting a photon with an energy exactly equal to the difference between the two specific energy levels.
• Minimum Wavelength ($\lambda_{\text{min}}$): There is a sharp "cutoff" at the short-wavelength end of the spectrum. This corresponds to an electron losing all its kinetic energy in a single collision to produce one high-energy photon.

Derivation of Minimum Wavelength

The maximum energy of a photon ($E_{\text{max}}$) is equal to the maximum kinetic energy of an electron ($eV$):

1. $E_{\text{max}} = eV$
2. Since $E = \frac{hc}{\lambda}$, then $E_{\text{max}} = \frac{hc}{\lambda_{\text{min}}}$
3. Equating the two: $eV = \frac{hc}{\lambda_{\text{min}}}$
4. Rearranging for the minimum wavelength: $\lambda_{\text{min}} = \frac{hc}{eV}$

Where:

• $h = 6.63 \times 10^{-34} \, \text{J s}$ (Planck’s constant)
• $c = 3.00 \times 10^8 \, \text{m s}^{-1}$ (Speed of light)
• $e = 1.60 \times 10^{-19} \, \text{C}$ (Elementary charge)
• $V = \text{Accelerating potential difference in Volts (V)}$

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2.3.2 X-ray Imaging and Contrast

A standard X-ray produces a 2D shadow image. The quality of this image depends on two main factors: Contrast and Sharpness.

Factors Affecting Contrast

Contrast allows us to distinguish between different types of tissue. It is determined by:

• The nature of the tissue: Materials with higher density and higher atomic numbers ($Z$) attenuate X-rays more effectively. Bone ($Z \approx 14$) appears white on a negative film, while soft tissue ($Z \approx 7$) appears darker.
• X-ray Hardness: "Hard" X-rays have higher energy (shorter $\lambda$) and are more penetrating, which actually reduces contrast. "Soft" X-rays have lower energy and provide better contrast but are more likely to be absorbed by the body, increasing the radiation dose to the patient.
• Contrast Media: When imaging soft tissues with similar attenuation coefficients (e.g., the intestines vs. surrounding muscle), a contrast medium like Barium or Iodine is used. These have high $Z$ values, making them highly opaque to X-rays.

Factors Affecting Sharpness

Sharpness is the definition of the edges. It is improved by:

• Reducing the area of the target (anode): A point source of X-rays reduces the "penumbra" (the blurred fringe at the edge of a shadow).
• Reducing the distance between the patient and the detector: This minimizes the spreading of the X-ray beam.
• Using a lead grid: This absorbs scattered X-rays that would otherwise hit the detector at angles and blur the image.

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2.3.3 Attenuation of X-rays in Matter

When a collimated beam of X-rays passes through a material, its intensity decreases exponentially with thickness.

The Attenuation Equation

$I = I_0 e^{-\mu x}$

• $I$: Transmitted intensity ($\text{W m}^{-2}$)
• $I_0$: Initial (incident) intensity ($\text{W m}^{-2}$)
• $\mu$: Linear attenuation coefficient ($\text{m}^{-1}$ or $\text{cm}^{-1}$)
• $x$: Thickness of the material ($\text{m}$ or $\text{cm}$)

Half-value Thickness ($x_{1/2}$)

The half-value thickness is the distance $x$ at which $I = \frac{1}{2} I_0$.

1. $\frac{1}{2} I_0 = I_0 e^{-\mu x_{1/2}}$
2. $0.5 = e^{-\mu x_{1/2}}$
3. $\ln(0.5) = -\mu x_{1/2}$
4. $-0.693 = -\mu x_{1/2}$
5. $x_{1/2} = \frac{\ln 2}{\mu} \approx \frac{0.693}{\mu}$

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2.3.4 Computed Tomography (CT) Scanning

Conventional X-rays suffer from the "overlap" problem: a 3D object is squashed into a 2D image, hiding depth information. CT scanning solves this.

The Step-by-Step Process:

1. Rotation: An X-ray tube and a bank of detectors rotate 360° around the patient.
2. Sectional Exposure: A thin, fan-shaped beam of X-rays is used to irradiate a single "slice" (section) of the patient from many different angles.
3. Data Collection: For each angle, the detectors measure the transmitted intensity, providing a 1D profile of attenuation.
4. 2D Reconstruction: A computer processes the intensity data from all angles of that specific slice. It uses complex algorithms to calculate the attenuation coefficient ($\mu$) for each small volume element (voxel) within the slice. This produces a 2D image of that section.
5. 3D Combination: The patient is moved slightly along the axis (z-axis), and the process is repeated for many slices.
6. Final Image: The computer stacks these 2D slices to build a high-resolution 3D digital model of the internal organs, which can be rotated and viewed from any direction.

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2.4 Worked Examples

Worked Example 1 — Minimum Wavelength Calculation

An X-ray tube operates at an accelerating potential of $95.0 \, \text{kV}$. Calculate the minimum wavelength of the X-rays produced.

Solution:

1. Identify variables: $V = 95.0 \times 10^3 \, \text{V}$.
2. State the equation: $\lambda_{\text{min}} = \frac{hc}{eV}$
3. Substitute values: $\lambda_{\text{min}} = \frac{(6.63 \times 10^{-34} \, \text{J s}) \times (3.00 \times 10^8 \, \text{m s}^{-1})}{(1.60 \times 10^{-19} \, \text{C}) \times (95.0 \times 10^3 \, \text{V})}$
4. Intermediate step: $\lambda_{\text{min}} = \frac{1.989 \times 10^{-25}}{1.52 \times 10^{-14}}$
5. Final Answer: $\lambda_{\text{min}} = 1.31 \times 10^{-11} \, \text{m}$

Worked Example 2 — Attenuation and Thickness

The linear attenuation coefficient of a specific muscle tissue is $0.45 \, \text{cm}^{-1}$. Determine the thickness of muscle required to reduce the incident X-ray intensity by $80\%$.

Solution:

1. Analyze the intensity: If intensity is reduced by $80\%$, then the transmitted intensity $I$ is $20\%$ of $I_0$. Therefore, $\frac{I}{I_0} = 0.20$.
2. State the equation: $I = I_0 e^{-\mu x} \Rightarrow \frac{I}{I_0} = e^{-\mu x}$
3. Substitute and solve for $x$: $0.20 = e^{-(0.45)x}$ $\ln(0.20) = -0.45x$ $-1.609 = -0.45x$ $x = \frac{-1.609}{-0.45} = 3.576...$
4. Final Answer: $x = 3.6 \, \text{cm}$ (2 s.f.)

Worked Example 3 — Half-Value Thickness

A beam of X-rays passes through a lead shield. It is found that $12 \, \text{mm}$ of lead reduces the intensity to $1/8$ of its original value. Calculate the half-value thickness ($x_{1/2}$) of lead.

Solution:

1. Conceptual approach: $1/8$ is $(1/2)^3$. This means the beam has been halved three times.
2. Calculate $x_{1/2}$: $3 \times x_{1/2} = 12 \, \text{mm}$ $x_{1/2} = 4.0 \, \text{mm}$
3. Alternative (Algebraic) approach: $\frac{1}{8} = e^{-\mu(12)}$ $\ln(0.125) = -12\mu \Rightarrow \mu = 0.1733 \, \text{mm}^{-1}$ $x_{1/2} = \frac{\ln 2}{0.1733} = 4.0 \, \text{mm}$
4. Final Answer: $x_{1/2} = 4.0 \, \text{mm}$

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2.5 Key Equations

Equation	Description	Data Sheet?
$\lambda_{\text{min}} = \frac{hc}{eV}$	Minimum wavelength of X-ray photons	No (Must derive)
$E = hf = \frac{hc}{\lambda}$	Energy of a single photon	Yes
$I = I_0 e^{-\mu x}$	Exponential attenuation of intensity	No (Must memorise)
$x_{1/2} = \frac{\ln 2}{\mu}$	Relationship between half-value thickness and $\mu$	No (Must derive)

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2.6 Common Mistakes to Avoid

• ❌ Wrong: Using $V$ in kilovolts (kV) directly in the $\lambda_{\text{min}}$ formula.
• ✓ Right: Always convert kV to V ($1 \, \text{kV} = 10^3 \, \text{V}$) to ensure SI unit consistency.
• ❌ Wrong: Confusing "reduced by 70%" with "reduced to 70%".
• ✓ Right: If reduced by 70%, then $I = 0.30 I_0$. If reduced to 70%, then $I = 0.70 I_0$.
• ❌ Wrong: Assuming $\mu$ is a constant for a material regardless of the X-ray energy.
• ✓ Right: $\mu$ decreases as X-ray energy increases (harder X-rays are more penetrating).
• ❌ Wrong: Describing a CT scan as a "3D X-ray" without explaining the mechanism.
• ✓ Right: Always mention that it involves taking multiple 2D images from different angles and using a computer to reconstruct the 3D image.
• ❌ Wrong: Using $\log_{10}$ instead of $\ln$ (natural log) when solving the attenuation equation.
• ✓ Right: The base of the exponential is $e$, so you must use the natural logarithm ($\ln$).

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2.7 Exam Tips

1. Derivation of $\lambda_{\text{min}}$: You are frequently asked to explain why there is a minimum wavelength. Use the phrase: "The minimum wavelength corresponds to an electron losing all its kinetic energy in a single collision to produce a single photon."
2. The Role of the Computer in CT: In CT scan questions, always emphasize that the computer is necessary to calculate the attenuation coefficients of the voxels and to combine the 2D slices into a 3D image.
3. Contrast vs. Sharpness: If a question asks how to improve an image, identify if the problem is visibility (Contrast) or blurriness (Sharpness).
   • To improve Contrast: Use a contrast medium or lower the X-ray tube voltage (softer X-rays).
   • To improve Sharpness: Use a smaller anode, a lead grid, or keep the patient closer to the detector.
4. Unit Consistency: The exponent $-\mu x$ must be dimensionless. If $\mu$ is in $\text{cm}^{-1}$, $x$ must be in $\text{cm}$. If $\mu$ is in $\text{m}^{-1}$, $x$ must be in $\text{m}$.
5. Logarithmic Graphs: If you plot $\ln(I)$ against $x$, the result is a straight line with a gradient of $-\mu$. This is a common way for examiners to present data.