1.4 - Photon Attenuation

## Principles of photon attenuation

Attenuation is the progressive loss of energy by a beam as it traverses matter.
A photon beam may be attenuated by any of the processes described in the previous section. There are some more useful concepts when considering the attenuation of photon beams.

### Exponential attenuation

If a beam is made up of a pure photon energy, attenuation occurs in an exponential fashion - ie. if 1 cm of material attenuates 50% of the photons in a beam, then 2 cm of a material will attenuate 75%. The formula is:

(1)
\begin{align} I(x)=\frac{I_0}{e^{\mu x}} \end{align}

Ie: The intensity of the beam (I) after passing through a material of thickness x is related to the original intensity of the beam divided by the natural number e to the power of the thickness x multipled by the linear attenuation coefficient mu.

### Non-exponential attenuation

If a beam is made up of a spectrum of energies (for example, in a beam produced by a typical linear accelerator or x-ray device) then attenuation will not be purely exponential. Lower energy photons are attenuated more rapidly and the beam hardens. Hardening refers to the loss of lower energy photons from the beam through attenuation. Hardening is useful as the low energy photons are frequently undesirable, as they contribute to scatter or surface dose. It can be performed by adding a filter to the device, that attenuates a significant portion of the low energy photons.

### Half Value Layer / Tenth Value Layer

The HVL is the thickness required to attenuate half of the original intensity of a beam, given by:

(2)
\begin{align} HVL = \frac{0.693}{\mu} \end{align}

The HVL is used to describe the penetration of superficial x-rays. For a bread with a broad spectrum of energies, the first HVL is often shorter as the lower energy photons in the beam are rapidly removed. The second or third HVL are generally closer in distance as the lower energy photons are already remvoed.

The TVL is the thickness required to attenuate the beam to one tenth of its original intensity.

(3)
\begin{align} TVL = \frac{log_e10}{\mu} \end{align}

The TVL is used in radiation protection to determine the number of thicknesses required for shielding or for beam blocks.

## Coefficients of Attenuation

### Linear Attenuation Coefficient

The linear attenuation coefficient, $\mu$, is a constant that describes the unique properties of a material that attenuates a photon beam. It has units of cm-1.

### Mass Attenuation Coefficient

The mass attenuation coefficient, $\frac{\mu}{\rho}$ is obtained by dividing the linear attenuation coefficient by the density of the material. The mass attenuation coefficient removes the density of the material from determining attenuation, instead basing attenuation off the atomic properties of the substance. It can be used instead of the linear attenuation coefficient:

(4)
\begin{align} I(x)=\frac{I_0}{e^{\frac{\mu}{\rho} \rho x}} \end{align}

### Energy Transfer Coefficient $\mu _{tr}$

It is useful to consider the amount of energy that is transferred by attenuated photons to electrons within the material. Most photons will interact through incoherent scattering, losing some energy and continuing through the material. Electrons are responsible for most of the dose deposition within the tissue. The energy transfer coefficient, $\mu _{tr}$, is found by multiplying the linear energy coefficient by the average energy transferred through photon interactions ($\bar{E_{tr}}$) and dividing this value by the photon energy $h \nu$.

(5)
\begin{align} \mu _{tr}=\frac{\bar{E_{tr}}}{h \nu}\mu \end{align}

### Energy Absorption Coefficient $\mu _{en}$

Electrons lose their energy in a material through collisional or radiative processes. Collisions (excitation / ionisation) occur when an electron interacts with orbital electrons, and usually deposit energy locally. Radiative events (bremsstrahlung) occur when electrons interact with the nucleus of an atom, generating an x-ray. These x-rays usually travel outside of the volume of interest. The average amount of energy lost through bremsstrahlung in a material is referred to as g, giving the formula for the energy absorption coefficient:

(6)
\begin{align} \mu _{en}=\mu _{tr}(1 - g) \end{align}

In tissues where atomic number is low or generated electrons have low energy, the value of g is very low. It becomes more important in materials such as lead which may give rise to a significant amount of bremsstrahlung.

## Calculation of mass attenuation and linear attenuation coefficient

The attenuation of photons within a material is due to the photon interactions described previously. Each interaction is dependent on the physical structure of the material and the energy of the photon. For a particular material, the total attenuation (linear attenuation coefficient) is due to the attenuation from each of these processes.

• Coherent scattering (Rayleigh) - $\sigma _{coh}$
• Photoelectric Effect - $\tau$
• Incoherent scattering (Compton) - $\sigma _{inc}$
• Pair and Triplet Production - $\pi$
• Photodisintegration

Giving:

(7)
\begin{align} \mu = \sigma _{coh} + \tau + \sigma _{inc} + \pi \end{align}

Given that photon interactions are dependent on the atomic properties of a material rather than its density, the attenuation coefficients for individual processes are often given as mass attenuation coefficients (divided by $\rho$):

(8)
\begin{align} \frac{\mu}{\rho} = \frac{\sigma _{coh}}{\rho} + \frac{\tau}{\rho} + \frac{\sigma _{inc}}{\rho} + \frac{\pi}{\rho} \end{align}

### Attenuation from Coherent Scattering

Coherent scattering is important for low kilovoltage photons, and increases with increasing atomic number.

### Attenuation from Photoelectric effect

The mass photoelectric attenuation coefficient is proportional to the cube of the atomic number (Z3) and inversely proportional to the cube of the beam energy (E3).

### Attenuation from Incoherent Scattering

The mass incoherent scattering attenuation coefficient is similar for most values of Z, but decreases slowly with increasing beam energy. It is most dependent on the electron density.

### Attenuation from Pair Production

Pair production only occurs with higher beam energies (over 1.02 MeV). The mass attenuation coefficient for pair production is linearly related to the atomic number. Increasing beam energy also increases the attenuation from pair production in a logarithmic fashion.