Interaction of X-Rays with Matter. X-ray fluorescence and absorption technique uses the interaction of x-rays with a material to determine its elemental composition and oxidation state. When the sample is irradiated with high-energy x-rays, on reaching a sample, some of the x-rays will be absorbed, and some will be scattered or the x-rays will be transmitted through the sample. If the atoms in the sample absorb X-ray energy by photoelectric absorption, this can cause subsequent characteristic x-ray emission or ejecting auger electron. X-ray Fluorescence (XRF) is the name given to the process of the emission of characteristic x-rays and forms the basis of XRF spectroscopy. The x-rays can also be scattered from the material. This scattering can occur both with and without loss of energy, called Compton and Coherent scattering respectively. These processes are discussed in the following sub sections more detail below.
3.2.1 Compton scattering
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This interaction is also known as , incoherent scattering or inelastic scattering. This process takes place when a photon interacts with a free electron in the outer shells of the atom (Figure 3-1). The result of this interaction is a deflected photon with a lower energy than the incident one and a recoil electron with energy transferred from the incident photon. Equation (3.1) represents the relationship between the scattered photon energy () and the incident photon energy () and the deflected angle of photon (θ):
where is the rest mass energy of the electron, which is considered to be at rest before the interaction. Kinetic energy of the recoil electron is maximized at θ =180° while the scatter photon retains most of its original energy when θ≈0°.
Figure 3-1 Compton Scattering
The Klein-Nishina (KN) differential scattering cross section per unit solid angle(3.2), describes the probability of a photon being incoherently scattered by a free electron at rest through an angle θ (Grieken and Markowicz 2002):
Where, is Klein-Nishina cross section for single free electron,
k is the photon energy in unit of rest mass,
ï± is the photon scattering angle, and
is the classical electron radius .
In the energy range of interest, the electron binding significantly alters the incoherent scattering cross section. The differential cross sections per solid angle(3.3) for incoherent scattering photon from an atom with atomic number Z are given as follows (Grieken and Markowicz 2002):
Where, is the incoherent scattering factor whose deviation from Z accounts for the binding correction and is the momentum transfer in unit (length)-1 and depends on the wavelength λ of the initial photon and the scattering angle θ and is given by.
3.2.2 Coherent scattering
This interaction also called elastic scattering or Rayleigh scattering. In this process the photon is scattered by a bound atomic electron; the result being coherently scattered photons that have the same energy and phase as an incident photon and the recoil atom is neither ionized nor excited as shown in the Figure 3-2:
Figure 3-2: Coherent scattering
The classic Thompson differential cross section describes the process of scattering between electromagnetic radiation and an electron, where the photon energy is considered to be much less than the rest mass of the electron and the target electron is free (Grieken and Markowicz 2002), by equation (3.4),
Where, is Thompson differential cross section per unit solid angle,
(θ) is the angle by which the photon is scattered, and;
is the classical electronic radius.
Modifications to the Thompson differential cross section were applied to include the electron distribution effect among one atom or molecule. The following expression (3.5) is used to describe the process between a photon and the total number of atomic electrons described by the Independent Atomic model IAM which is valid for momentum transfer >5nm-1. This model assumes that there are no inter-molecular interference effects and so is the theoretical form factor for free atoms.
Where, is the form factor which express the strength of the scattering signal of a photon from a material and accounts for the accounts for the atomic number, and the momentum transfer.
Another modification includes the oscillatory structure function which describes the interatomic and molecular bond effects. These effects are dominant at low momentum transfer values. Whether the atomic or the molecular form factor is used to calculate the coherent cross section is based on the condition of the experiment (including the incident photon energy and the angle of scattering) and the approximation assumed for dominant process accounts for scattering.
3.2.3 Photoelectric Absorption
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At the energy utilized in this research, the photoelectric absorption is the main interaction of interest, which forms the basic theory behind x-ray fluorescence and absorption technique.
In the photoelectric absorption process, the incident photon is completely absorbed and an energetic photoelectron is ejected by the atom from one of its shells (Figure 3-3). The K shell is the most probable origin of the photoelectron. The energy of the photoelectron is given by(3.6)-
Where, is the photoelectron energy,is the incident photon energy, andis the binding energy of the photoelectron in its original shell. The incidence photon energy needs to be at least equal to the binding energy of the photoelectron for the process to take place (Knoll G.F. 2000).
There are two main mechanisms of which an atom de-excites following a photoelectric absorption event as illustrated below
The photoelectric absorption process creates a vacancy as a result of ejection of an inner shell electron. This vacancy may be filled by an electron from higher level orbits and is accompanied by emission of characteristic x-ray line. A Kα photon is produced when the vacancy originates in the K shell and is subsequently filled by an electron from the L shell See Figure ().
Figure3-3: Photoelectric absorption followed by Kα Characteristic X-ray emission.
The Kα photon energy is equal to the difference in binding energy between the K and L shells. Emission of L, M, N…series characteristic X-rays occurs in the same manner when vacancies of these outer shells are filled. The energy of the electron energy levels is characteristic of the atom. Thus the fluorescent x-ray is characteristic of the atom in which the fluorescence process is occurring. More details about characteristics x-ray lines and transition are elaborated ... in section 3-5.
An alternative process to the emission of a characteristic x-ray photon, as a result of the photoelectric absorption process, is the ejection of one of the outer shell electrons (Auger electron). In this process the excitation energy of the atom is transferred directly to that electron. The energy of the Auger electron equals the difference between the original atomic excitation energy and the binding energy of the shell from which the electron is ejected. Auger electron emission is dominant for elements with Z<31 and the probability of its emission increases as the differences between the corresponding energy levels decrease (Figure-4).
Figure 3-4: Photoelectric absorption followed by Auger electron emission
3.2.4 Fluorescent yield
As explained above, not all the incident x-rays result in fluorescence. The fluorescent yield expresses the fraction of all cases in which the atom de-excites by emission of a characteristic x-ray.
Fluorescent yield of the K shell, is given by
Where, is the total number of characteristic x-rays from the atom and is the number of primary K shell vacancies. The fluorescent yield decreases as the target atomic number decreases (Knoll G.F. 2000).
3.3 Total Attenuation Coefficient
The incident photon beam experiences different interactions with the sample and each type of interaction has a corresponding cross-section, which determines the probability of each type of interaction (Knoll G.F. 2000). Assuming narrow beam geometry (point source, monoenergetic source, and collimated photon beam), the attenuation of the beam as a result of all competing processes while passing through a material is described by the following equation:
Where, Ð† is the transmitted intensity, is the incident intensity, τ the absorber thickness in cm, is the total linear attenuation coefficient which represents the fraction of photons that interact per unit thickness of the absorbing material and accounts for all possible interactions for a given photon energy :
Where, is the linear attenuation coefficient for photoelectric effect which its dependence on the atomic number of the material Z and the energy of the initial photon for low atomic material can be given as , is the linear attenuation coefficient for coherent scattering (), and is the linear attenuation coefficient for Compton scattering, which decreases with increasing the incident photon energy and is independent of Z of the absorber material ().
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The linear attenuation coefficient depends on the density of the material, in addition to its dependence on the photon energy and the atomic number of the absorber. The mass attenuation coefficient (cm2/gm) is the ratio between the linear attenuation coefficient (cm-1) and the density of the material (gm/cm3). This parameter was introduced to avoid obtaining different coefficient values of the same material in different phases. The mass attenuation coefficient values can be calculated using the computer program XCOM (Berger and Hubbell 1998). These values are arranged and tabulated, and can be used to identify any element or compound of interest. Figure (3-5) shows the,, and for soft tissue at different energies.
Figure 3-5: Mass attenuations coefficients for soft tissue
3.3.1 The X-ray Absorption Coefficient
In this research, X-ray absorption spectroscopy (XAS), which measures the absorption of x rays as a function of the incident x-ray energy, was utilised. At the energy used in this research for the XAS measurements (>6 and < 11 keV), as the Compton and coherent effect can be neglected at this energy range, the total attenuation coefficient becomes equivalent to the absorption coefficient, as is shown in figure (3-6). Hence, the absorption coefficient has been of primary concern in this case and can be given as:
Again Ð† is the transmitted intensity,is the incident intensity, τ the absorber thickness in cm,gives the probability that x-rays will be absorbed by photoelectric interaction. The depends on x-ray energy (E), atomic number (Z), the density (), and atomic mass A, and for low Z martial is given as:
The dependence ofon both Z and E is the main property which makes x-rays absorption useful for medical imaging techniques.
3.3.2 Absorption Edge
In photoelectric absorption, the photon disappears and the electron is ejected from an atom. An electron cannot be ejected if the incident photon energy is lower than the binding energy of the electron shell. When the incident x-ray has energy equal to that of the binding energy level of an electron, there is a sharp rise in the x-ray absorption coefficient known as an absorption edge, which corresponds to the characteristic level energies of the atom. (Figure 3-7) depicts an example of an absorption edge spectrum. The main features of the spectrum are the decrease of intensity with energy, the sharp edge at the characteristic energy levels of each atom, and the series of oscillatory structure above the edge as shown in the enlarged view of the edge.
Figure 3-7: K Absorption edge of Fe
3.4 Electronic transitions and characteristic x-ray emissions
The Pauli's exclusion principle states that no two electrons can exist in an atom in the same state which means that no two electrons in an atom can have same quantum number. The four quantum numbers which specify the state of an electron are principle quantum number (n), orbital angular momentum (l), magnetic orbital quantum number (ml) and magnetic spin quantum number. These quantum numbers can take specific values only for instance in X-Ray spectral notation the group for level, l can take values from 0 to 1, ml can take values from -l to l and the spin quantum number ms can take only two values +1/2 and -1/2. The total angular momentum of an atom is given by J and is summation of orbital angular momentum and magnetic spin quantum number (J = l ± ½). There are a certain number of allowed transitions between electronic shells that are limited by the selection rules for magnetic dipole set by quantum physics as:, is the total orbital angular momentum of an atom and is the total angular momentum of the atom. The values of is represented using small letters for = 0,1,2,3,4 and 5 in modern spectral notation.
Figure 3-8 shows the x-ray energy level diagram showing some of the transitions giving rise to characteristic x-ray lines. The figure also shows the x-ray, optical and spectral notations.
Figure3-8: Electronic transitions and characteristic x-ray emissions
According to the selection transition rule, transitions can take place between and , and , and and . However, under normal conditions transitions are forbidden between lower and higher orbital for an instance transition cannot take place between and, and or and. The above selection rule accounts for the transitions that give rise to the more intense lines of x-ray spectral series. However, small fainter lines, forbidden by the selection rules have been observed. The 26 electrons of Fe in the normal state are designated as follow: given that the prefix to the notation is the number and the superscripts are the number of electrons.Fe has electrons in the 4th shell N while M shell is incomplete, which is in accordance with the chemical properties of the atom. The atom of Fe can be raised from normal state to higher energy state by absorbing radiant energy or bombarding them by electrons. When the atom is in higher energy state it is described as excited state. When the atoms return to lower energy state, radiation is emitted. The spectrum of Fe (II) or Fe (III) shows spectral lines with energy equal to the transition between