Introduction to Photon Attenuation
The discussion above has been concerned with the interaction of photons with individual atoms, but it is also necessary to consider the macroscopic behavior of photons traversing matter. For this purpose, linear and mass attenuation coefficients are used, which are directly related to the total cross-section.
Photons may undergo more than one interaction as they pass through bulk material. For example, an initial scatter interaction might be followed by a second scattering process, which could then be followed by further scattering, photoelectric absorption, or no further interactions, with the photon exiting the bulk material.
Linear Attenuation Coefficient
Consider a thin uniform slab of material of thickness \( dx \), irradiated with a beam of photons incident normally on the slab. Individual photons may pass through the slab without interacting, or they may be absorbed or scattered. The probability that an individual photon will interact in this thin section is given by:
\( \sigma N_a dx \)
Where \( N_a \) is the number of interaction centers (atoms) per unit volume and \( \sigma \) is the total interaction cross-section per atom. The quantity \( N_a \sigma \) is known as the linear attenuation coefficient, denoted by \( \mu \).
The linear attenuation coefficient \( \mu \) can be expressed as:
\[ \mu = \frac{N_a \sigma}{\rho} \]
Where \( N_a \) is the number of interaction centers, \( \sigma \) is the total cross-section, and \( \rho \) is the density of the material.
Exponential Attenuation
Now consider a thick slab of material, and let \( \Phi(x) \) represent the fluence of photons that have not interacted after passing through thickness \( x \). The expected change, \( d\Phi \), in this fluence after passage through a further thickness \( dx \) is given by:
d\Phi = -\mu \Phi dx
This equation describes the exponential attenuation of a photon beam, also known as Beer’s law. The relationship between the fluence at depth \( x \) and the initial fluence \( \Phi_0 \) is given by:
\Phi(x) = \Phi_0 e^{-\mu x}
Mass Attenuation Coefficient
The linear attenuation coefficient \( \mu \) depends on the density of the material, which can vary based on its physical state. Therefore, the mass attenuation coefficient \( \frac{\mu}{\rho} \), which is independent of density, is more commonly used in data compilations. This quantity is typically expressed in units of \( \text{cm}^2/\text{g} \) in radiological studies.
Energy Transfer and Absorption
For dosimetric purposes, the energy transferred to secondary electrons resulting from photon interactions is essential. The linear energy transfer coefficient \( \mu_{tr} \) and mass energy transfer coefficient \( \mu_{tr}/\rho \) allow for the calculation of this energy. The energy transferred to secondary electrons is given by:
\mu_{tr} = \langle T \rangle / h \nu
For photons of energy \( h \nu \) traversing a distance \( dx \) in a material, the energy transferred to secondary electrons is:
dT = \mu_{tr} \Phi dx
The kerma (kinetic energy released per unit mass) can then be calculated as:
K = \frac{\mu_{tr}}{\rho} \Phi
Mass Energy Absorption Coefficient
Some of the energy transferred to secondary charged particles is lost to radiative processes, primarily bremsstrahlung. To account for this, the mass energy absorption coefficient \( \mu_{en}/\rho \) is used:
\mu_{en}/\rho = \mu_{tr}/\rho \cdot (1 - g)
Where \( g \) is the energy fraction lost to radiative processes. For diagnostic energies, \( g \) is typically zero.
Contribution of Interactions to Total Mass Attenuation Coefficient
The total mass attenuation coefficient is the sum of the individual mass attenuation coefficients for each interaction type (e.g., photoelectric absorption, Compton scattering, pair production, etc.). The contribution of each interaction depends on photon energy and material properties:
\mu_{\text{total}} = \mu_{\text{photoelectric}} + \mu_{\text{Compton}} + \mu_{\text{pair production}} + \mu_{\text{Rayleigh scattering}}
Coefficients for Compounds and Mixtures
The mass attenuation coefficients for compounds or mixtures can be obtained by a weighted summation of the coefficients of the constituent elements:
\frac{\mu}{\rho} = \sum_i w_i \frac{\mu_i}{\rho_i}
Note: \( w_i \) represents the normalized weight fractions of the elements \( i \) (or mixture components) present in the material. For mixtures, radiative losses of secondary electrons can also be considered.