Interactions with Matter Notes

Interactions with Matter

Interactions Processes

Excitation & Ionization: Optical radiation, K x-rays, and filament electrons interact with absorber materials, leading to phenomena such as Bremsstrahlung (photon emission due to electron deceleration) and delta rays (energetic electrons that cause further ionization).
Transmission of X-rays: Beam attenuation as X-rays pass through absorbers. The intensity of the transmitted beam decreases exponentially with the thickness of the material. $I(x) = I_0e^{-\mu x}$ , where:
- $I(x)$ is the intensity of the transmitted beam at thickness $x$ .
- $I_0$ is the initial intensity of the X-ray beam.
- $\mu$ is the linear attenuation coefficient of the material, dependent on energy and atomic number.
- $x$ is the thickness of the absorbing material.
Half Value Layer (HVL): The thickness of absorbing material required to reduce the intensity of the X-ray beam by half. It is a measure of beam quality or penetrability. HVL is only truly representative for monoenergetic beams. For polyenergetic beams, the 1st and 2nd HVLs will differ due to beam hardening (preferential removal of lower energy photons).
kVp: Kilovolt Peak, the peak voltage applied to the X-ray tube, which accelerates electrons from the cathode to the anode.
keV: Kiloelectron-volts, a unit of energy, often describing the energy of resultant X-rays. The kVp value represents the maximum possible keV of the X-ray photons produced.

Probability of an Interaction

The probability of an interaction between X-rays and matter depends on:
- Number of electrons per unit volume (electron density, $\rho$ ) in the material.
- Atomic number (number of protons, Z) of the atoms in the material.
These factors are functions of the absorbing material's composition (e.g., tissue type, X-ray target material, shielding material).
The type of interaction (coherent scattering, photoelectric absorption, Compton scattering) also depends on the energy (E) of the incident radiation.

Significance of Interactions

X-rays in diagnostic imaging are essentially shadows created by the differential attenuation of the primary X-ray beam.
The depth and characteristics of the shadow depend on the strength and type of interaction between X-rays and the tissues.
Attenuation can either provide or degrade contrast in the image. Different tissues attenuate the primary beam differently based on their composition and the type of interaction that predominates at a given energy.

Coherent Scattering (Rayleigh Scattering)

Coherent scattering accounts for approximately 5-10% of all X-ray interactions in diagnostic radiology and remains relatively constant across different energies.
Involves an interaction between a photon and an electron where the photon's energy is lower than the binding energy (BE) of the electron, thus no ejection occurs. Instead, the atom is temporarily excited.
The electron temporarily stores the energy and then releases it as a photon with the same energy and wavelength as the initial photon but in a different direction.
No energy absorption occurs during coherent scattering, so there is no dose delivered to the patient, just scatter that can slightly degrade image quality.
Coherent scattering is not a major interaction effect at MRS (Magnetic Resonance Spectroscopy) energies.
The likelihood of coherent scattering interaction is proportional to: $P_{coherent} \propto \frac{Z}{E^2}$ , where Z is the atomic number and E is the energy of the incident photon.
Steps:
1. An incident photon interacts with an electron, typically in one of the outer shells of the atom.
2. The electron temporarily stores the energy from the photon.
3. After a brief period, the electron releases the stored energy as a photon, which travels in a different direction.
Factors:
- Atomic Number (Z) of absorbing material: As Z increases, Coherent Scattering increases (Z ⬆, CS ⬆).
- Energy (E) of incoming X-ray: As E increases, Coherent Scattering decreases (E ⬆, CS ⬇⬇).

Photoelectric Absorption

In photoelectric absorption, an incident photon transfers all of its energy to an inner-shell electron of an atom. This interaction is the basis for radiographic contrast.
A portion of the photon's energy is used to overcome the electron's binding energy and eject it from the atom as a photoelectron.
The remaining energy is transferred to the electron as kinetic energy (KE). Thus, $KE{electron} = E{photon} - BE_{electron}$ .
The photoelectron rapidly loses its kinetic energy through collisions and travels only a short distance from its original location.
Photoelectric absorption leads to electron interactions and has potential biological effects due to energy deposition.
A vacancy in one of the inner electron shells is created, making the atom unstable. An electron from an outer shell drops down to fill the vacancy, releasing energy as either a characteristic X-ray photon or an Auger electron.
The energy of the characteristic radiation is specific to the atom and depends on its atomic number (Z) and the binding energies of its electron shells. High Z materials produce X-rays, while low Z materials may produce light or very low-energy photons.
Steps:
1. An X-ray photon interacts with an inner-shell electron of an atom.
2. The X-ray is absorbed by the electron, and the electron is ejected from the atom as a photoelectron. If the electron is from the K-shell, its binding energy is denoted as $BEK$ . The photoelectron moves away with kinetic energy equal to $Ei - BEK$ , where $Ei$ is the incoming photon's energy.
3. The ejected electron leaves a vacancy in the inner shell, leaving the atom in an excited state.
4. Electrons from higher energy levels transition to fill the vacancy, resulting in the emission of characteristic radiation or Auger electrons to stabilize the atom.
Photoelectric absorption is most likely when the incident photon energy is slightly higher than the binding energy of an electron shell.
The probability of photoelectric absorption decreases rapidly with increasing photon energy.
Inner shell electrons are primarily involved in photoelectric absorption at X-ray energies.
The probability of photoelectric absorption is proportional to $P_{photoelectric} \propto \frac{Z^3}{E^3}$ .
Key dependencies:
- Energy (E) of incoming X-ray: As E increases, Photoelectric effect decreases (E ⬆, PE ⬇).
- Atomic Number (Z) of absorbing material: As Z increases, Photoelectric effect increases (Z ⬆, PE ⬆).
If the energy of the photon is too small (less than BE), no absorption occurs. If the energy is too large (high keV-MeV), a different interaction process, such as pair production, becomes more likely.
At low energies, photoelectric absorption mainly involves outer electrons. As energy increases, inner electrons become more involved.
The probability of photoelectric absorption decreases smoothly with energy until the energy is just above the binding energy of an electron shell. At this point, there is a sharp increase in absorption called an absorption edge. For example, the K absorption edge occurs when the photon energy is equal to the binding energy of the K-shell electrons.
Contrast agents, such as barium and iodine, are chosen with specific K-edges to maximize X-ray absorption and enhance contrast in imaging. For example, Barium has $BE_K \approx 37.4$ keV.
Materials with specific K-edges are used to provide high energy filtration, removing lower-energy photons to reduce patient dose and beam hardening artifacts.
Rare earth materials with appropriate K-edges are used in lightweight protective aprons as a substitute for lead.
Materials with specific K-edges are also used for beam shaping, which is the selective filtration of X-rays to improve image quality by reducing Compton scatter and patient dose.
The probability of photoelectric absorption increases significantly with the atomic number (Z) of the absorber:
- Materials with higher atomic numbers have more electrons, increasing the likelihood of interaction.
- The probability of interaction is proportional to $(Z)^3$ .

Differential Absorption in Diagnostic Radiography

Differential absorption, the difference in X-ray absorption between different tissues, is crucial for creating diagnostic images.
This process allows us to visualize different tissues and structures in the body.
Tissues differ in atomic number and density, resulting in variations in X-ray absorption.
The photoelectric effect is a primary contributor to differential absorption because of its strong dependence on atomic number.
Large Atomic Number Ratios:
- The greater the ratio between Z numbers of two tissues $\frac{ZA}{ZB}$ , the easier it is to distinguish between them.
- Example: comparing bone and soft tissue;
- $Z{bone} \approx 14$ and $Z{soft tissue} \approx 7$ , so $\frac{Z{bone}}{Z{soft tissue}} = \frac{14}{7} = 2$
- The absorption ratio is proportional to $(\frac{ZA}{ZB})^3 = (2)^3 = 8$
- Therefore, bone absorbs approximately 8 times more radiation than soft tissue at certain diagnostic energies, primarily due to the photoelectric effect.
- This difference in absorption is why bones appear white on radiographs.
Small Atomic Number Ratios:
- Imaging tissues with similar atomic numbers $\frac{ZA}{ZB} \approx 1$ (e.g., breast tissue) requires careful technique.
- To distinguish between materials, differences in the photoelectric effect must be maximized, which can be achieved by manipulating the energy of the X-ray beam.
- Since $(\frac{ZA}{ZB})^3 \approx (1)^3 \approx 1$ , small variations are amplified by using lower X-ray energies. This increases the probability of photoelectric absorption.
- At lower energies, even small differences in tissue composition lead to significant variations in absorption.
Mammography is conducted at low energies to image breast tissue effectively, as breast cancers have similar atomic numbers to surrounding dense breast tissue.
Reducing energy is feasible only when imaging thin structures like the breast, as lower energy X-rays have less penetration power and are more easily attenuated. This is why tissue compression is necessary in mammography to reduce thickness and improve image quality.
For thicker structures, higher X-ray energies are needed to ensure that the X-rays penetrate the structure and reach the imaging plate. This reduces overall photoelectric absorption, particularly in lighter atoms such as those in soft tissue.
- Example: Chest radiography is performed at high energies (120 kVp) to ensure adequate penetration of the lungs and mediastinum.
- However, higher X-ray energies also introduce more Compton scattering, which degrades image quality.

Compton Scattering (Incoherent Scattering)

Compton scattering is the most common interaction process in diagnostic radiology but is often the least desirable due to its negative impact on image quality.
It involves a collision between an incident photon and a loosely bound (free) electron, resulting in both absorption and scattering. A free electron is one in which the binding energy is small compared to the energy of the incident photon, typically an outer-shell electron.
Compton scattering is significant in the energy range of approximately 30 keV to 30 MeV.
The incident X-ray photon loses some of its energy to the electron and is deflected in a different direction with lower energy.
This scattering process leads to the generation of secondary radiation, as the absorber becomes a source of scattered photons and energetic electrons.
The scattered X-ray photon can:
- Leave the patient, contributing to radiation exposure to personnel and potentially reaching the image receptor as noise.
- Undergo further Compton scattering events within the patient.
- Undergo photoelectric absorption if its energy is sufficiently reduced.
- The freed electrons contribute to patient dose through ionization and excitation of nearby atoms.
Factors determining Compton probability:
- Electron Density of Material: As electron density (number of electrons per unit volume) increases, the probability of Compton scattering increases. More electrons mean more targets for interaction.
- Energy of Photon: As photon energy increases, the probability of Compton scattering decreases, but this decrease is slower compared to the photoelectric effect. Higher energy photons are more likely to undergo Compton scattering than photoelectric absorption. Compton scattering dominates in many diagnostic radiology (DR) and nuclear medicine (NM) energy ranges. A reduction in Compton scattering occurs only after 1.02 MeV, when pair production becomes significant.
- $P_{Compton} \propto \frac{electron \, density}{energy \, of \, radiation}$
Electron and Physical Density:
- As physical density increases, the number of electrons per volume also increases, leading to a higher probability of Compton interaction.
- Most elements have approximately equal numbers of protons and neutrons; thus, the number of electrons is roughly proportional to the mass density.
- Absorption is similar in all materials if their physical density is the same, assuming the energy range is such that Compton scattering dominates.
- Because Compton scattering involves outer-shell electrons, the binding energy is a negligible factor, and the atomic number (Z) is not a primary determinant.
The angle of the scattered X-ray photon is primarily determined by the energy of the incident X-ray photon.
- At low incident X-ray energies, the photon can be scattered at any angle, with approximately equal probabilities of forward and back scatter.
- As incident energy increases:
- The scatter angle decreases, resulting in more forward scatter.
- The scattered X-ray photon retains more energy.
- The Compton electron receives less energy.
Lower energy initial photons lead to large angles of deflection. These photons exit the patient and may interact with individuals nearby, increasing the risk of radiation exposure. Lower energies also increase the probability of photoelectric absorption and associated biological effects.
Higher energy initial photons result in low angles of deflection. High-energy scattered photons in the forward direction significantly degrade image quality by adding noise to the image.
The Angle of Deflection & Energy of Scattered Radiation are related by the Compton shift formula:
- $\Delta \lambda = 0.024(1 - cos \theta)$
- Where $\Delta \lambda$ is the change in wavelength between the initial and scattered photon, and $\theta$ is the scattering angle.
- At larger scattering angles:
- There is a larger change in wavelength ( $\theta \uparrow : Cos \theta \downarrow : (1 - Cos \theta) \uparrow : \Delta \lambda \uparrow$ ), meaning the scattered photon has less energy.
- When $\theta \approx 180^\circ$ , the scattered photon energy is at its minimum, and the Compton electron receives its maximum energy.
The relationship between energy and wavelength is expressed as: $E = \frac{hc}{\lambda}$
In summary, as incident energy increases:
- The number and energy of scattered photons in the forward direction increase.
- The overall number of Compton scattering events decreases as more photons are transmitted through the patient without interaction.
- The percentage of attenuation events due to Compton scattering increases up to 1.02 MeV and then decreases as pair production becomes more prevalent.

Summary of Interactions

Coherent Scatter
- Photon in – Photon out
- Negligible effects on image quality and dose
- $P \propto \frac{Z}{E^2}$
Photoelectric Effect
- Photon in – Electron out (Plus characteristic radiation and/or Auger electrons)
- Provides radiographic contrast
- $P \propto \frac{Z^3}{E^3}$
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