The Beer Lambert Law

Published Date: 02 Nov 2017

Attenuation is the measure of incident radiation dispersed or photons "lost" as they propagate through a medium [Ball3]. Attenuation is a collective term for all of the factors that contribute to the loss of transmission of a signal through a medium. The main factors are photoelectric absorption, scattering processes and pair production. Each has a finite probability of occurring that is dependent on the energy of the incident radiation and the atomic properties of the medium that the radiation penetrates. The summation of these probabilities yields the linear attenuation coefficient for a given material, dependent on the energy of the penetrating radiation. As an example, Figure [fig:Linear-Attenuation-Coefficient] shows the linear attenuation coefficient for water. Each physical process that contributes to the coefficient is also plotted to show which processes are dominant across what ranges of energy. Ross filters operate in the energy range dominated by photoelectric absorption. It is of interest at this point to note that Figure [fig:Linear-Attenuation-Coefficient] shows that Compton scattering is the dominant attenuation process over the energy range the TLD and filter stack will observe, and pair production is the dominant process over the energy range the activation layers will observe.

From investigating what factors contribute to how probable attenuation is and why the probability of each varies with energy, the criteria for which materials are suitable for each component can be determined. Throughout the following sub-sections, the source of the photons being attenuated is assumed to be a laser-induced plasma unless otherwise stated.

Photoelectric absorption is the process in which an incident photon is completely absorbed by an electron. In its place a photoelectron is ejected from the atom. It is the dominant process for low energy X-rays, as can be seen from Figure [fig:Linear-Attenuation-Coefficient]. As described by the shell model of the atom, when an electron absorbs the incident photon's energy it can be excited to a more energetic state. All of the photon's energy is absorbed meaning that the photon ceases to exist due to the conservation of energy. If however, the energy of the absorbed photon is higher than that of the binding energy of the electron to its shell, the electron can be ejected from the atom, carrying with it most of the absorbed photon's energy [Knoll_Photoelec]. The energy of the photoelectron is given by:

where E_{b}=

binding energy of the electron to its shell and \hbar\omega=

energy of the incident photon. X-rays have an energy range between 0.1 KeV - 1 MeV [X-ray Energies], which is a sufficient amount of energy to eject an electron from the most tightly bound shell of an atom; the K shell. The K shell is therefore the most probable origin of a photoelectron, since the X-rays are too energetic to be absorbed by an electron further away from the nucleus. This is because electrons that are further away from the nucleus experience a reduction in the Coulomb force felt between the two, and so the electron's binding energy is reduced.

As a result of an electron being liberated from the K shell, there exists a vacancy in the shell. As described by the Aufbau principle, electrons occupy the lowest energy state possible, and so an electron from another shell decays to fill this vacancy, emitting an X-ray (to conserve energy) of characteristic energy. These secondary X-rays are usually absorbed close to their origin via photoelectric absorption, but is is possible for them to either scatter off of another atom or to escape the medium entirely. A secondary source of photons such as this will affect the measurement of the X-ray radiation from the plasma since their wavelengths depend on the atom and the energy level they originated from. They are not related to the X-rays emitted by the plasma.

A rough approximation of the probability of photoelectric absorption occuring per atom within a medium is given by [Photo_Prob]:

where n

varies between 4 and 5 depending on the photon energy range of interest [Evans]. As can be deduced from the equation, photoelectric absorption is more probable in high Z materials, and at lower photon energies (as is shown to be true in Figure [fig:Linear-Attenuation-Coefficient]).

With regards to the Ross filters, to achieve the aim of maximising the transmission of low energy X-rays emitted from a plasma, low Z materials should be used. As a point of interest, this relationship between absorption and Z number also shows why dense materials are used for radiation shielding, since they strongly absorb radiation.

7.2 K - Edges

A photon must have an energy greater than or equal to the binding energy of an electron to liberate it from its shell. If the photon has insufficient energy to be absorbed, then the photon propagates through the medium unaffected by photoelectric absorption. As the energy of the photon increases towards that of the binding energy of an electron to a shell, the probability of the photon being transmitted unattenuated increases. However, if the energy of the photon is equal to the binding energy of the electron, the photon is absorbed, resulting in a drastic reduction in transmittance. The photon effectively disappears and a photoelectron is generated in its place. This phenomenon is known as an absorption edge, or K-edge if the absorption occurs within the K Shell [Ball2, Williams]. As the photon energy increases further past the absorption edge, the probability that the photon is attenuated by photoelectric absorption decreases and so the transmittance tends toward a maximum of 1. Figure [fig:LinAttAl] shows how the attenuation coefficient abruptly increases when the photon energy is equal to that of the binding energy of an electron to the K shell. This increase in attenuation corresponds to a decrease in transmittance.

For low Z elements such as aluminium, X-rays are too energetic to be absorbed by an electron occupying a higher energy state such as the L or M shells, and hence those with a lower binding energy to the atom. However, for high Z elements it is possible to observe L and M edges since more energy is required to eject an electron from these shells. This is due to the increased Coulomb attraction between the electrons and the nuclei since the nuclei consists of more protons in high Z elements. Figure [fig:LinAttPd] demonstrates this effect.

The energy at which absorption edges occur is characteristic of each element and is closely linked to the production of characteristic X-rays. Though absorption edges may seem undesirable due to their negative effect on the probability that a photon is transmitted, K-edges are key to how Ross filters work and to how the energy limits of each filter are defined.

7.3 Compton Scattering

As the energy of the incident photons is further increased, Compton scattering becomes the dominating factor in contributing to the attenuation of the photons. When a photon is incident upon an electron, it absorbs some of the photon's energy and is ejected from the atom. The more energy that is imparted on the electron (relative to the electron's binding energy), the faster it will recoil and thus the less energy the scattered photon will have. This is due to conservation of energy. The angles at which the recoil electron and the scattered photon travel after the collision must satisfy the conservation of momentum. The maximum amount of energy that can be imparted on an electron is dependent on the maximum energy of the photon. As any energy up to this maximum can be imparted, both the electron and the photon can scatter off at any angle. The energy of the scattered photon is given by [Knoll_Photoelec]:

where \hbar\omega

is the energy of the incident photon, m_{0}c^{2}

is the rest mass-energy of the electron and \theta

is the angle of the scattered photon. As can be seen from Figure [fig:Linear-Attenuation-Coefficient], the effect of Compton scattering on attenuation increases with increasing energy, but then at higher energies it decreases towards zero. This is because high energy photons have a tendency to scatter forwards, as predicted by the Klein-Nishina formula [Knoll_Photoelec]. This reduces the amount of attenuation, since photons which scatter forwards appear to be unattenuated.

The probability of Compton scattering occuring per unit length of a medium is given by [Evans]:

where N

is the amount of atoms in the medium and \sigma_{e}

is a function of the incident photon's energy. From this expression, it can be inferred that the more atoms that are present (ie. the thicker the material) and the higher the Z number, the higher the probability that Compton scattering will occur. This makes sense, as both higher N

and higher Z

means an increased amount of electrons for the photons to potentially scatter off of as they propagate through the material.

Compton scattering is the main process which will affect the amount of radiation entering through the second component; the TLD stack, dependent on what filters are utilised to obtain the desired low energy cut-offs. As can be inferred from Eqn ([eq:Compton]), to obtain a suitable high low-energy cut-off either a thick low Z filter or a thinner high Z filter can be utilised to a similar effect.

Rayleigh scattering is a coherent scattering process in which a photon scatters off of an electron without any loss of energy. When a photon is absorbed by an electron, the electron excites to a higher energy state. For the electron to obey the Aufbau principle and tend towards the lowest energy state possible, the electron can immediately emit a photon of the same energy as the incident photon to decay back to its original energy state. Even though there is no loss of energy in this interaction, there is a possibility that the emitted photon will have a different direction of propagation, and thus increasing the attenuation of the incident radiation.

As demonstrated by Figure [fig:Co-Scat], Rayleigh scattering is more dominant at low X-ray energies. Its effects are more pronounced in high Z materials, since there are more electrons of which the photons can interact with. However, Rayleigh scattering is always dominated by photoelectric absorption and so it will be neglected from discussions on attenuation.

If the energy of the incident photons exceed 1.022 MeV, which is twice the rest mass of an electron, pair production becomes feasible. A photon now has enough energy to penetrate deep into the atom, interacting directly with the Coulomb field of the nucleus. When this occurs, the photon can be separated into an electron and a positron, both carrying energy of 511 KeV. A positron is the antimatter version of an electron; it has the same rest mass but it is of opposite charge. It is worth mentioning that this process can occur within the Coulomb field of an electron, but since the field is of much smaller magnitude the chances that it does is unlikely. Since there is a loss in the amount of photons transmitted through the material, attenuation increases.

However, this is not the main reason why pair production must be taken into consideration. When a positron produced by pair production interacts with an electron, they annihilate [Evans]. The result of this is the production of two 511 KeV photons to conserve energy, emitted in opposite directions to conserve momentum. This annihilation radiation will then be detected by the diagnostic, which is undesirable since the annihilation radiation contains no information about the incident radiation. The annihilation radiation will occur within the energy range of the TLD stacks, and so in the design of this component there must be a way to account for the unwanted radiation.

There is no simple equation to model the probability of pair production occuring, but the value varies with approximately Z^{2}

of the absorbing material [Evans]. With this approximation in mind, it can inferred that pair production is most probable for incident radiation of energy >> 1.022 MeV and for high Z materials.

By summing the probabilities of each process that contributes to attenuation, the linear attenuation coefficient can be determined:

This is a measure of the probability that radiation penetrating a medium will be attenuated, and is true for a narrow beam of incident radiation [Ball2]. This statement allows the linear attenuation coefficient to be utilised in calculations to determine the materials to be used as the diagnostic's components. This is due to the spectral bins each component creates, and so the heterogeneous X-ray emission from the plasma can be modelled as many homogeneous components of radiation.

A more useful attenuation coefficient is the mass attenuation coefficient, which takes into account the density of the medium attenuating the incident X-rays. For example, ice is less dense than water but they would both have the same linear attenuation coefficient. However they would have different mass attenuation coefficients. The mass attenuation coefficient is given by:

where \rho

is the density of the medium. Again using the example of water (Figure [fig:Linear-Attenuation-Coefficient]), the mass attenuation coefficient is a sum of the mass attenuation coefficients for hydrogen and oxygen, since both have different Z. When the medium consists of a compound or a mixture of elements, the mass attenuation coefficient can be determined from [Knoll_Photoelec]:

where w_{i}

is the weight fraction of element i

within the compound c

. This equation will be utilised when investigating methods of improving the spectra of Ross filter pairs for the first component and more importantly, the low energy cut-offs of each TLD within the TLD stack for the second component.

The Beer-Lambert law determines the relationship between incident radiation and the degree of attenuation it experiences as it propagates through a medium. It is thus the most important equation for the following work on Ross filters and the TLD stack. The intensity of the transmitted radiation is defined as [Knoll_Photoelec]:

where I_{0}=

intensity of the incident radiation (before the medium) and t=

thickness of the medium. If the intensity is normalised and the mass attenuation coefficient is substituted for the linear attenuation coefficient:

where T=

percentage of radiation transmitted. The Beer-Lambert law is no longer dependent on knowing the intensity of the incident radiation. As can be seen from the equation, the amount of photons unattenuated and transmitted through the material exponentially decays if the mass attenuation coefficient and the thickness increase.

The final factor to consider in how attenuation affects the transmittance of radiation through a medium is the possibility of build-up effects. Photons which are scattered off of atoms will scatter off of other atoms after the initial interaction, and so there is a chance that this scattered radiation can find it's way to a detector measuring transmittance. This attenuated radiation is hence undesirable since it does not contain any information about the incident X-rays and would have to be taken into account if the material was thick and there were lots of targets to scatter of off. However, for the thicknesses of materials used to create the diagnostic's components, build-up effects can be neglected [PC RC].