Photon Interactions with Matter

1.6.1. Exponential Absorption of Photon Beam in Absorber

The primary parameter for describing the penetration of X-ray or γ-ray photon beams into an absorbing medium is the linear attenuation coefficient, denoted as \( \mu \). The attenuation coefficient \( \mu \) depends on the energy of the photon (\( h\nu \)) and the atomic number \( Z \) of the absorber. It represents the probability per unit path length that a photon will interact with the absorber.

The attenuation coefficient \( \mu \) is typically determined experimentally by directing a narrow, collimated monoenergetic photon beam of energy \( h\nu \) onto a suitable detector. An absorber of varying thickness \( x \) is placed between the photon source and the detector, and the change in the intensity of the beam is measured. The intensity of the beam is reduced by the absorber, and this reduction is related to the thickness of the absorber and the properties of the material.

Mathematical Formulation of Attenuation

The rate of change in the intensity \( I(x) \) of the photon beam as it passes through an absorber of thickness \( x \) can be described by the differential equation:

\( \frac{dI(x)}{dx} = -\mu I(x) \quad \text{(1.41)} \)

Here, the negative sign indicates that the intensity decreases with increasing absorber thickness. Alternatively, the fractional reduction in intensity, \( -\frac{dI}{I} \), can be expressed as:

\( -\frac{dI}{I} = \mu \, dx \quad \text{(1.42)} \)

Both equations describe the same physical process, and equation (1.42) can be directly integrated to give the total attenuation of the photon beam. The integration over the thickness \( x \) of the absorber from 0 to \( x \) and the intensity from \( I(0) \) (no absorber) to \( I(x) \) (with absorber) results in:

\( \int_0^x \frac{dI}{I} = - \int_0^x \mu \, dx \)

Which simplifies to the exponential attenuation law:

\( I(x) = I(0) e^{-\mu x} \quad \text{(1.44)} \)

Where:

The equation shows that the intensity of the photon beam decreases exponentially as it passes through the absorbing material. This exponential attenuation is a key principle in understanding how materials absorb or scatter photons, and it is crucial for various applications in radiation shielding, medical imaging, and nuclear physics.

Example Calculation: Photon Attenuation

Suppose we have a photon beam with an initial intensity \( I(0) = 100 \, \text{units} \), passing through a material with a linear attenuation coefficient \( \mu = 0.2 \, \text{cm}^{-1} \). We want to calculate the intensity of the beam after passing through a material of thickness \( x = 5 \, \text{cm} \).

Using the exponential attenuation formula \( I(x) = I(0) e^{-\mu x} \), we can substitute the known values:

\( I(5) = 100 e^{-0.2 \times 5} = 100 e^{-1} \approx 36.79 \, \text{units} \)

So, after passing through 5 cm of the material, the intensity of the photon beam is approximately \( 36.79 \, \text{units} \).

Important Notes

Note: The attenuation coefficient \( \mu \) depends on the photon energy and the atomic number of the material. Higher atomic number materials and higher photon energies generally lead to different values of \( \mu \). Additionally, at higher photon energies, the attenuation coefficient may decrease due to the dominance of Compton scattering and pair production processes.

Photon Interactions with Matter - Characteristic Absorber Thicknesses

Characteristic Absorber Thicknesses

Equation (1.44) represents the standard expression for the exponential attenuation of a monoenergetic narrow photon beam passing through an absorber. The intensity of the photon beam decreases exponentially with increasing absorber thickness \(x\), as shown in the graph below (Fig. 1.4), which is a typical exponential plot of intensity \(I(x)\) versus absorber thickness \(x\).

There are three special absorber thicknesses that are commonly used to characterize photon beams. These are the **Half-Value Layer (HVL)**, **Mean Free Path (MFP)**, and **Tenth-Value Layer (TVL)**, each providing insight into the beam's penetration capability through a given material.

1. Half-Value Layer (HVL)

The **Half-Value Layer (HVL)**, denoted \( x_{1/2} \), is defined as the thickness of the absorber required to reduce the intensity of the photon beam to half of its original intensity. In other words, \( I(x_{1/2}) = 0.5 I(0) \).

The relationship between the HVL \( x_{1/2} \) and the attenuation coefficient \( \mu \) can be derived as follows:

\( I(x_{1/2}) = I(0) e^{-\mu x_{1/2}} = 0.5 I(0) \)

Solving for \( x_{1/2} \), we obtain:

\( \mu x_{1/2} = \ln 2 \)

Thus, the HVL is given by:

\( x_{1/2} = \frac{\ln 2}{\mu} \approx \frac{0.693}{\mu} \quad \text{(1.46)} \)

2. Mean Free Path (MFP)

The **Mean Free Path (MFP)**, denoted \( x_e \), is the average distance that a photon travels through the absorber before undergoing an interaction. It is the absorber thickness at which the intensity of the photon beam is reduced to \( 1/e = 0.368 \) of its original intensity, i.e., \( I(x_e) = 0.368 I(0) \).

The relationship between the MFP \( x_e \) and the attenuation coefficient \( \mu \) can be derived as follows:

\( I(x_e) = I(0) e^{-\mu x_e} = 0.368 I(0) \)

Solving for \( x_e \), we obtain:

\( \mu x_e = 1 \)

Thus, the MFP is given by:

\( x_e = \frac{1}{\mu} \quad \text{(1.48)} \)

3. Tenth-Value Layer (TVL)

The **Tenth-Value Layer (TVL)**, denoted \( x_{1/10} \), is the thickness of the absorber that reduces the intensity of the photon beam to one tenth (10%) of its original intensity. In other words, \( I(x_{1/10}) = 0.1 I(0) \).

The relationship between the TVL \( x_{1/10} \) and the attenuation coefficient \( \mu \) can be derived as follows:

\( I(x_{1/10}) = I(0) e^{-\mu x_{1/10}} = 0.1 I(0) \)

Solving for \( x_{1/10} \), we obtain:

\( \mu x_{1/10} = \ln 10 \)

Thus, the TVL is given by:

\( x_{1/10} = \frac{\ln 10}{\mu} \approx \frac{2.303}{\mu} \quad \text{(1.50)} \)

Relationship Between Characteristic Thicknesses

From the above equations, we can express the linear attenuation coefficient \( \mu \) in terms of each characteristic thickness. These relationships are as follows:

\(\mu = \frac{\ln 2}{x_{1/2}} = \frac{\ln 10}{x_{1/10}} = \frac{1}{x_e} \quad \text{(1.51)}\)

From these equations, we can also derive the relationships between the characteristic thicknesses:

\(\frac{x_{1/2}}{x_{1/10}} = \frac{\ln 2}{\ln 10} \approx 0.301\)

Thus, we obtain the following general relationships among the characteristic thicknesses:

\( x_{1/2} \approx 0.301 x_{1/10} \quad \text{and} \quad x_e = \frac{x_{1/2}}{\ln 2} \)

Important Notes

Note: These characteristic thicknesses—HVL, MFP, and TVL—are essential in radiation shielding calculations, medical imaging, and understanding photon transport in various materials. The larger the values of these thicknesses, the more material is needed to attenuate the photon beam. The HVL and TVL are particularly useful in evaluating the effectiveness of shielding materials.