Photoelectric absorption

An alternative process of interaction of X-rays with matter, apart from scattering, is the so-called photoelectric absorption. It allowed to popularise X-rays in medical diagnostics in the form of X-ray projectional radiography, being currently replaced by alternative, less-irradiating imaging techniques. In the photoelectric absorption, an incident X-ray photon is fully absorbed by an atom and its energy is transferred to an electron, which is ejected from the atom. The interaction leaves the atom ionised. A measure of absorption is given by the linear absorption coefficient µ, which can in principle be determined in a transmission experiment, in which the ratio between intensities of the incident and the outgoing beams are measured. The intensity of X-ray beam traversing through the specimen of a thickness ∆zfalls off exponentially, according to the Lambert-Beer law [21].

Therefore, the transmitted intensity, I_out(z), is given by the following expression:

Iout(z) = Iine^−µ∆z (1.2.25) where I_in is the intensity of the incident X-ray beam. The linear absorption coefficient µ is an element-specific quantity and a function of the X-ray energy. It exhibits a strong dependence on the atomic number of the element, varying approximately as Z⁴. Further-more, it decreases with an increasing incident photon energy, being inversely proportional to its third power, E⁻³. Additionally, the linear absorption coefficient is related to the absorption cross-section per atom, σ_ab, through the following formula:

µ= ρ_mN_A

A σ_ab (1.2.26)

whereρ_m andA are the mass density and the molar mass of the atom, respectively, while N_A is the Avogadro constant. As photoelectric absorption cannot be explained in terms

1.2 Interaction of X-rays with matter 39

Fig. 1.2.6: Absorption and scattering cross-sections as functions of the incident photon energy for an atom of carbon (C), calcium (Ca), and iron (Fe). The sharp, discountinuous jumps in the absorption cross-sections correspond to the respective absorption edges. The scattering cross-sections represent a sum of coherent and incoherent scattering contribu-tions. Data retrieved from [25].

of classical physics, the quantum-mechanical formalism must be invoked to consider the photon-electron interaction in the context of the first-order perturbation theory. Through the Fermi’s Golden Rule [21], which determines the transition probability of such a system, it is possible to obtain the exact absorption cross-section values.

It is though more common to quantify photoelectric absorption in terms of the mass absorption coefficient, defined as µ/ρ_m. At a fixed photon energy, µ/ρ_m is constant for a given element, independently of the form of matter. Therefore, the mass absorption coefficient of a chemical compound or a mixture is approximately a weighted average of the coefficients for the constituent elements, as follows:

µ ρm

_mix

=^X

j

w_j µ ρm

!

j

(1.2.27) where w_j is the weight fraction of the jth element. Energy-dependent mass absorption coefficients of first 92 elements are tabulated in [25]. Fig. 1.2.6 shows absorption (solid lines) and total scattering (dashed lines) cross-sections as functions of the incident photon energy for an atom of carbon (green), calcium (blue), and iron (red). The total scattering cross-sections represent sums of coherent and incoherent scattering processes. The ab-sorption cross-section lines feature discontinuous and sharp jumps at the element-specific energies. These sharp increases are called absorption edges and are a direct consequence of discrete binding energies of atomic electrons. As an example, the K-shell electrons in Fe have a binding energy of 7.112 keV. If the incident photon energy exceeds that value, pho-toelectric absorption may result in recoil of a K electron and annihilation of the photon.

Yet, in case the photon energy decreases below the threshold of 7.112 keV, this particular absorption process becomes energetically impossible making the absorption cross-section fall sharply by a certain amount (K absorption edge). In the lower photon energy range

40 X-ray radiation characteristics (<1 keV), there are other discontinuities, as in the case of Ca and Fe in Fig. 1.2.6. These are the L absorption edges indicating energy thresholds required for the removal of L-shell electrons. The apparent 3-level structure of the L edges stems from the degeneracy of the L-shell electron energies. It is caused by: (1) screening of the nuclear charge by the inner K-shell electrons, resulting in lower energy of the 2s electrons than the 2p electrons, and (2) spin-orbit coupling (splitting of the 2p level). As demonstrated in Fig. 1.2.6, the photoelectric absorption usually dominates over other processes (coherent and incoher-ent scattering), which also contribute to an overall attenuation of the X-ray beam. The scattering processes prevail for the lightest elements (see the plots for a carbon atom) at higher photon energies.

As mentioned before, when an atom absorbs an incident X-ray photon, the photon energy is transferred to one of the core-shell electrons, as illustrated in Fig. 1.2.7A. As a result, the electron is expelled to the continuum, leaving the atom in an excited state.

The ionised atom will relax back into its ground state, by filling the inner-shell vacancy with an outer-shell electron. Since the inner-shell electrons are more strongly bound to the nucleus, the resulting energy difference will then be emitted in one of the two forms:

(1) fluorescence radiation (Fig. 1.2.7B) or (2) an Auger electron (Fig. 1.2.7C).

In the first process, as shown in Fig. 1.2.7B, the energy excess is emitted as X-ray fluorescence photons. It is discrete and monochromatic characteristic radiation which can be used to identify the atom it was emitted from. First to observe this was Henry Moseley, who discovered a systematic relation between the energy of the most intense characteristic line, currently known as Kα, and the atomic number Z of a given element. He reflected this in the following empirical law [21]:

E_Kα [keV]≈1.017×10⁻²(Z−1)² (1.2.28) It can be explained by the fact, that the differences between the binding energies of the re-spective atomic orbitals are strictly dependent on the atomic numberZ. Additionally, the possible transitions of electronic states within an atom are defined by the selection rules for electric dipole radiation [26]. Considering the timescale of such transitions, relaxation of an outer-shell electron to the inner-shell hole remaining after the recoil photoelectron

K L M continuum

A

X-ra y photon

K L M continuum

B

Kβ

K^α

K L M continuum

C

Fig. 1.2.7: The photoelectric absorption (A) results in ejection of a photoelectron into the continuum and leaves the atom ionised. The atom may relax back into its ground state following two phenomena: (B) emission of a characteristic fluorescent X-ray radiation or (C) ejection of an outer-shell Auger electron.

1.2 Interaction of X-rays with matter 41 occurs of the order of 10 to 100 fs. From the Heisenberg’s uncertainty principle, the nat-ural width of the resulting characteristic X-ray lines is therefore of the order of 0.01 eV, with more precise values depending on the considered element and transition. Summaris-ing, the energy values of fluorescent lines are tabulated for the entire table of elements and can be found in, e.g. the X-ray Data Booklet [27].

The second, competitive relaxation process subsequent to the photoelectric absorption is shown in Fig. 1.2.7C. It is non-radiative and involves transferring the energy excess into another, loosely-bound electron, whose binding energy is less than the transferred energy.

Consequently, the electron is ejected from the atom. The Auger process dominates over the fluorescence emission for the low-Z elements. The quantity that measures probability of relaxation occurring through the fluorescence emission rather than through the Auger process is called the fluorescence yield. It is particularly low for all biologically-relevant structural elements (C, N, O), for which the Auger process prevails, and increases with the atomic numberZ.