2
Basis of XAFS

2.1 Interactions of X‐Rays With Matter

When electromagnetic waves of x‐ray energies (>120 eV, wavelength < 10 nm) interact with the electron clouds of a material, three principal effects can dominate the result (Figure 2.1).

1. There is elastic (Rayleigh) scattering with the incoming and outgoing photons having the same energy. Diffraction is a specific case of this for crystalline samples.
2. Additionally, there is inelastic (Compton) scattering. In this case the outgoing photon is of lower energy (longer wavelength). This is similar to Raman spectroscopy, which generally employs photons in the visible or near infrared to probe molecular vibrations or rotations. Using x‐rays as the excitations, the energy transfer can be to vibrations and to electronic transitions, both core and valence. Hence this can include components of XAFS spectroscopy.
3. For x‐ray absorption spectroscopy the most important type of event is absorption. This photoelectric effect involves a transition of an electron in a core orbital creating an excited state. This highly excited state can relax by emitting energetic electrons and photons of longer wavelength (fluorescence).

Figure 2.1 Interaction of x‐rays with materials.

2.1.1 Absorption Coefficients

As shown in Figure 2.1, absorption attenuates the x‐ray flux transmitted through a sample, which follows an exponential reduction by:

(2.1)

where I₀ and I_t are the incident and transmitted intensities, respectively, μ is the linear absorption coefficient, and l the sample length. The mass absorption coefficients, μ_m (=μ/ρ, where ρ = the density of the material) are tabulated for all elements[1–3] and thus the absorption characteristics of any material can be calculated by a mass‐weighted summation of the absorption coefficients of the elements at a given energy, according to equation 2.2.

(2.2)

The absorption properties of elements vary greatly with atomic number. Higher Z elements have more electrons at deeper potentials and thus have stronger interactions with incoming x‐rays. As the energy of x‐rays increase then the interaction becomes weaker and transmission increases through a sample. This is a strong effect related approximately to the λ³. The effect can be seen in Figure 2.2. This shows that the absorption of a solution of a salt of the tungsten oxo‐anion in a solvent of light elements decreases very significantly even over the energy range of a single XAFS spectrum.

Figure 2.2 W L₃ XAS of (NBu₄)₂[WO₄] (10 mM) in CH₃CN.

(Source: Diamond, B18, data from Richard Ilsley)

2.1.2 Absorption Edges

As is evident in Figure 2.2, the drop in absorption coefficient is arrested by a sharp rise in absorption, termed the absorption edge. This occurs when the photon energy corresponds to the binding energy of a core orbital. Absorption edges are conventionally labeled alphabetically according to the principal quantum number of the electron shell of the element being studied: K for n = 1, L for n = 2, M, for n = 3, and so on. Hence the label, K, corresponds unambiguously to a transition from a 1 s orbital. For the second shell, different energies would be anticipated for excitation of the 2 s and 2p orbitals. Accordingly, the next highest absorption edge energy after the K is attributed to the 2 s orbital, and is termed variously the L(I), L_I or L₁ edge. However, rather than being a single absorption edge resulting from the 2p electrons there are two—the L₂ and L₃ edges.

The removal of an electron by photo‐ionization creates an electron vacancy (a core hole) leaving a 2p⁵ sub‐shell. Coupling between the orbital angular momentum, l (1 for a p orbital), and the electron spin, m_s (½), gives rise to two j states: 2p_3/2 and 2p_1/2. In accord with Hund’s rules, the former, corresponding to j = l + s, should be the more stable ion with the sub‐shell being more than half full. So the third highest energy edge will be the L₂ with the ion having a 2p_1/2 sub‐shell and the fourth will be the L₃ edge affording a 2p_3/2 configuration. Figure 2.2 is showing the L₃ edge of tungsten, with an energy near 10207 eV. At higher energies lie the K edge at 69525 eV, the L₁ at 12100 eV, and the L₂ at 11544 eV, just past the end of the spectrum in Figure 2.2. These values show the large difference in energy between the electrons in the 1 s and 2 s orbitals, and a smaller difference, but still substantial, between 2 s and 2p. Rather less obvious, perhaps, is the large spin‐orbit coupling energy giving a 1300 eV difference between the L₂ and L₃ edges. This is due to the strong relativistic effects of these core electrons, which have very high kinetic energies. This absorption edge was studied for an experiment in preference to the L₂ edge for two reasons.

1. It is the more intense. This is due to the higher degeneracy of the j = 3/2 state as compared to the ½ (4 versus 2).
2. From the energy of the L₃ edge up to that of the L₂ all features in the spectrum are unambiguously associated with the L₃ transition. After the L₂ edge, there will be an overlap of features from the two edges. Due to the large energy difference between these two particular edges, this effect will be small, but that might not be the case for the corresponding edges of molybdenum, the element above tungsten in Group 6, which differ by about 100 eV (L₃ 2520 eV, L₂ 2625 eV).

The same principle applies to edges based upon 3d orbitals. There will be two j states generated—3d_3/2 and 3d_5/2—with the latter being the more less stable and thus requiring less energy to create. This is summarized in Table 2.1.

Table 2.1 Absorption edges and their corresponding electron configurations.

Absorption edge	Electron excited	Excited states
K	1 s	1 s1/2
L1	2 s	2 s1/2
L2, L3	2p	2p1/2, 2p/3/2
M1	3 s	3 s1/2
M2, M3	3p	3p1/2, 3p/3/2
M4, M5	3d	3d3/2, 3d/5/2

Tables of x‐ray absorption energies are readily available, not the least from the “X‐ray Data Booklet.”[3]

2.1.3 XANES and EXAFS

As indicated above, the absorption edge is associated with transitions of electrons in core orbitals to vacant states. Two types of vacant state can be envisaged: i) to a vacant valence orbital and ii) leaving the absorbing atom as a wave into the continuum (Figure 2.3). The first of these (Figure 2.3a) will affect the features observed near the absorption edge threshold, including pre‐edge features. The virtual valence orbitals accepting the electron promoted from the core may be either largely localized on the absorbing atom, or one that is very substantially delocalized onto neighboring atoms, and thus have charge transfer characteristics. In Figure 2.3b, the x‐ray is depicted as having sufficient energy to escape from the absorbing atom, and thus have an excess kinetic energy. This kinetic energy provides the photoelectron with its momentum and wavelength.

Figure 2.3 Transitions involved in a) XANES and b) EXAFS spectrum features.

2.1.3.1 XANES Features

The features in the XANES region then are due to transitions to unoccupied (virtual) states. The most probable pathway will involve dipole (Laporte) allowed transitions, which follow the selection rule Δl = ±1. Thus for K absorption edges, where the ground state electron is in the 1 s orbital (l = 0), it is the density of the empty p states that is most important. This will also hold for the less studied L₁ edge, which involves transitions from the 2 s orbital. For the L₃ edge, transitions to the both s and d states are formally allowed, but those to the d states have a substantially higher probability. The effects of this are illustrated in Figure 2.4, which compares the L₃ and L₁ edges of a tetrahedral ion with a 5d⁰ valence configuration.

Figure 2.4 The XANES regions at the a) L₃ and b) L₁ absorption edge of (NBu₄)₂[WO₄] (inset) (10 mM) in CH₃CN solution (10 mM) recorded in transmission.

(Source: Diamond, B18, data from Richard Ilsley)

As is evident, there is a strongly allowed absorption at the absorption edge at the L₃ edge. This can be ascribed to a dipole allowed transition of an electron from the 2p orbital to the 5d states of tungsten. All of these are vacant providing a high probability for the transitions, which, in this anion, are symmetry allowed. The lower energy set of these, of e symmetry in the tetrahedral point group T_d, (Figure 2.5a) displays considerable 5d character, but there is much 5d‐6p mixing evident in the t₂ set (Figure 2.5b). However, at the L₁ edge, an allowed transition from the 2 s orbital (a₁ symmetry) would be have t₂ symmetry, as in that shown in Figure 2.5b. There is a prominent absorption on the rising slope of the absorption edge. This arises from the significant degree of allowed‐ness in this transition, but the probability is not as high as for the totally allowed 2p to 5d transitions as seen in the L₃ edge.

Figure 2.5 Low‐lying vacant states as calculated for [WO₄]²⁻ in aqueous solution using Spartan’16 using the ωB97X‐D/6‐31G* method. a) one of the e set and b) one of the t₂ set.

The hybridization effect would be expected to be dependent upon the geometry and symmetry at the absorbing atom. In Figure 2.6, the K edge XANES features of a selection of chromium centers are presented. Here the tetrahedral complex displays an intense first absorption that is significantly below the energy of the absorption edge (14 eV), forming this very strong pre‐edge feature. This Cr(VI), complex will also have a valence configuration with the d‐p hydridized orbital set (t₂) unfilled. One of these triply degenerate orbitals is illustrated in Figure 2.7; the p‐d mixing is high in this case, with the p character predominating. Also shown in this figure is one of the vacant triplet of β‐spin states associated with the configuration of [Cr(OH₂)₆]³⁺. In a centrosymmetric site, p‐d mixing is symmetry forbidden, and thus the low‐lying empty states are essentially 3d in character. Thus transitions from a 1 s orbital to one of these states will require a different mechanism such as distortion by coupling of the transition with an antisymmetric vibration of the metal center (e.g., the t_1u antisymmetric O‐Cr‐O stretch) or via a higher order (quadrupole) transition. Both of these effects will be of lower probability and the resulting transitions will be weaker. Hence, the octahedral centers in [Cr(CO)₆] () and Cr₂O₃ (), which both also have vacant 3d levels of different symmetry to that of the vacant 4p orbitals (t_1u) display weaker pre‐edge features.

Figure 2.6 Cr K edge XANES of Cr metal, [Cr^o(CO)₆], Cr^III₂O₃ and K₂[Cr^VIO₄].

(Source: Data courtesy of Sofia Diaz‐Moreno, Roberto Boada‐Romero, and Luke Keenan, Diamond, I20)

Figure 2.7 Low‐lying vacant states as calculated for two chromium complexes in aqueous solution using Spartan’16 using the ωB97X‐D/6‐31G* method. a) [CrO₄]²⁻ and b) [Cr(OH₂)₆]³⁺.

The edge positions of Cr metal, Cr^III₂O₃, and K₂[Cr^VIO₄] follow a trend with oxidation state: an increase in Z_eff with oxidation state, which will lower the energy of the 1 s core orbital and will increase the ionization potential (Section 2.1.3.2). However, [Cr⁰(CO)₆] exhibits an absorption edge energy very similar to that of the Cr^III oxide. This demonstrates that the position of the absorption edge is also a function of the ligand set, and indeed upon the coordination geometry, particularly for soft‐donor ligands.[4]

Near to, and above, the energy of the absorption edge the excited electron has attained the Fermi energy and can move within the material. At low kinetic energies (<10 eV) the mean free path of the photoelectron in solids is relatively long (1 to 10 nm) (Figure 2.8). It falls to a minimum of about 0.5 nm at 50–100 eV, and only rising above 1 nm again above 1000 eV. Hence, in the XANES region, the photoelectron can interrogate and be scattered by structural arrays of a significant number of shells. In principle these states might be calculated and discussed in a quantum mechanical way; the large scale of the slab of the structure that may need to be considered can make this challenging (Section 5.3.2). In the XANES region, the photoelectron electron has such a long mean free path that the scattering is from a large number of atoms—called multiple scattering. XANES features can then be challenging to model and interpret and so have more often been used in a finger print mode in comparison with reference materials of known structure.

Figure 2.8 Typical electron mean free path in a material—modeled with silicon.

2.1.3.2 Edge Position

The energy of the absorption edge itself had been identified as chemically dependent early in the history of x‐ray absorption spectroscopy (Section 1.2). The apparent position of the absorption edge can be seen to occur at different energies in Figure 2.6.

There have been two alternative definitions used for measuring the edge position:

1. The energy of the absorption edge at 50% of the edge jump and
2. The energy of maximum slope (first derivative) on the absorption edge.

The latter is the more commonly employed measurement point. It is a directly observable point whereas the edge jump needs to be estimated by a backward extrapolation from the post‐edge spectrum. The maximum slope is close to, but a few eV above, the Fermi energy of a metal. It thus provides a starting estimation of the threshold energy of the photoelectron, E_0.

The absorption edge energy then may be related to the binding energy of photo‐excited electron, which will be affected by the Z_eff (Effective Atomic Number) of the binding site. If Z_eff is increased, for example, by increasing the oxidation number of the absorbing atom, the overall charge on a complex or increasing the electronegativity of neighboring atoms, then the energy levels of the core orbitals would be lowered, thus increasing their binding energies (Figure 2.9). In an excited state, with the electron transferred into an unoccupied valence orbital, then an increased Z_eff will also increase the binding energy of the destination orbital. The edge shift will be the difference between these two values, hence the edge position will increase with Z_eff if the effect on the core orbital dominates. If the transition is largely charge transfer in nature, then the binding energy of the acceptor orbital would not be expected to change much with the effective atomic number of the absorbing atom, and in this case the edge position will certainly be dominated by the ground state effects.

Figure 2.9 Energy level diagrams showing the effect of increasing the Z_eff from that of the reference material (center): a) when the transition is localized on the absorbing atom and b) if there is significant charge transfer.

Experimental evidence for the usefulness of the edge position can be seen from sulfur K edge spectra (Figure 2.10). The apparent edge position varies by about 10 eV between sulfide, S²⁻, and sulfate(VI), SO₄²⁻. This follows the direction expected if the effect to be dominated by the shift in the binding energy of the core orbital. The 1 s orbital is relatively shallow in energy and screening by the 3 s and 3p valance electrons of sulfur would be expected to cause a significant effect. So differentiation of S(‐II), S(0), S(IV), and S(VI) is viable from such measurements. Interesting too is the increase in the height of the first transition from a 3s²3p⁶ configuration through to 3s⁰3p⁰. The increase in p vacancies (holes) (n_h) enhances the intensity of the dipole allowed 1 s to 3p transition. For both [SO₄]²⁻ and [SO₃]²⁻ the acceptor orbitals are largely of S 3p character (Figure 2.11) and are S‐O antibonding in nature; for sulfate(VI), this is triply degenerate as compared to being doubly degenerate for sulfate(IV). On that ground alone, the pre‐edge feature would be anticipated to be more intense for [SO₄]²⁻. The intensity of both these features masks the onset of the continuum states normally considered to represent the absorption edge.

Figure 2.10 Sulfur K edge XANES of a series of compounds of different oxidation state.

(Source: Recorded on Lucia when at the Swiss Light Source, data from Michal Perdjon‐Abel)

Figure 2.11 Low‐lying vacant states as calculated for two sulfate ions in aqueous solution using Spartan’16 using the ωB97X‐D/6‐31G* method. a) [SO₄]²⁻ and b) [SO₃]²⁻.

However, as we can imagine from the spectra shown in the figures above, the measured‐edge position is a function of the perceived Z_eff, the coordination geometry, and the valence electron configuration. So correlations between edge position and oxidation number must be carried out judiciously. As is evident from the chromium examples in Figure 2.6, the nature of the ligand as well as the oxidation state and coordination geometry can have a significant influence.

2.1.3.3 The EXAFS Effect

From Chapter 1, we can see that less intense, broader oscillations than those in the XANES region can extend for 100 s of eVs beyond the absorption edge. In the energy range of 50 to 1000 eV, the mean free path of the photoelectron is in the range of 5–10 Å. The cause of the oscillation should then be local, and it is described by back‐scattering by the electron clouds from neighboring atoms (Figure 2.12).

Figure 2.12 a) The photoelectron outgoing wave and b) with back scattering from a neighboring atom.

The photoelectron behaves as a wave the wavelength (λ). This wavelength is related to its momentum (p) by the de Broglie equation λ = h/p, where h is Planck’s constant. The momentum is also related to the photo‐electron wave vector, k, by p = ħk, with ħ = h/2π. Since kinetic energy and momentum are defined as E = ½mv² and p = m_ev, the following relationships emerge for an electron of mass m_e: 2Em_e = p² and thus ħk = √(2Em_e). At a particular X‐ray photon energy E above the threshold energy Eo, the photoelectron wave vector, of units Å⁻¹, will be given by:

(2.3)

(2.4)

The electron wave has the wavelength, λ = 2π/k. The effect of the backscattered wave is that when it is in phase with the photoelectron wave the probability of absorption increases; it is decreased when these waves are out of phase. The result is that the absorption has a sinusoidal relationship to the wave vector k.

EXAFS datasets are based upon fractional changes in absorption as a function of k, χ(k). This fraction is the deviation between the observed absorption jump above E₀ from the expected jump if there were no EXAFS features. This can be written as:

(2.5)

As a result, χ(k) requires identifying E₀ from the observed spectrum, extrapolating the absorption through edge as if the edge were not there (pre‐edge background), and identifying the absorption post edge as if there were no EXAFS (post‐edge background). The background edge jump is the difference between the two backgrounds, μ₀(k). The values μ(k) are from the difference between the pre‐edge background and the observed absorption. The steps required are illustrated in Figure 2.13.

Figure 2.13 Steps to identify EXAFS, χ(k). a) Experimental x‐ray absorption spectrum of the L₃ edge of (NBu₄)₂[WO₄] in MeCN solution, showing pre‐edge background and post‐edge background, b) the resulting χ(k), c) the k² weighted EXAFS, k².χ(k) and d) the magnitude (solid), real part (dashed) and imaginary part (dotted) of the Fourier transform of k².χ(k).

(Source: Diamond, B18, data from Richard Ilsley)

As can be seen, the EXAFS oscillations decay quickly with increasing k value. In this example, the back‐scattering is from a single type of oxygen atoms within the tetrahedral [WO₄]²⁻ ion. In order to show the oscillations more clearly across the spectrum, the XAFS amplitudes that are viewed and analyzed are generally multiplied by kⁿ, where n is generally 1 to 3; k² weighting illustrated in Figure 2.13. The Fourier transform of the last curve shows a dominant Fourier component in real space (R in Å) with a distance similar to, but shorter than, the expected value for the W‐O bond length.

2.1.3.4 EXAFS Quantification

The local scattering model proposed by Stern was a key to understanding the application of EXAFS in the analysis (Section 1.4). The model was based upon a single scattering process, in other words the photoelectron is considered as being scattered back from one atom, thus traveling 2R_j, for an interatomic distance R_j. The photoelectron wave is modeled as a plane wave interacting with small atoms, approximations that hold more closely in the EXAFS region with photoelectron energies of more than 50 eV. This was known to be a practical approach. As will be discussed in Chapter 5, full analysis relaxes these approximations with the electron wave considered to be a sum of spherical harmonics and higher order scattering, in which the photo‐electron visits two or more atoms before its return to the central atom. Equation 2.6, which applies to a randomly oriented sample like a powder, glass or solution, provides a good basis to introduce the factors important in EXAFS.

(2.6)

The observed EXAFS, χ(k), is a summation of the back‐scattering for a series of shells each composed of the same atom within a relatively narrow band of distances. Experimentally, one would like to determine the interatomic distance between the central absorbing atom and the scattering atom (R_j). The sinusoidal term is dominated by this factor and thus the oscillations provide a good measure of the interatomic distance. The second important information element is the number of atoms in that shell, N_j.The amplitudes of the EXAFS features are linearly related to the magnitude of the coordination number. The third piece of information would be to identify the element within that shell. This does not appear to be within equation 2.6 at first sight, but there are two terms that bear the characteristics of the back‐scattering element. One is the back‐scattering amplitude f(k). This provides the shape envelop of the back‐scattering as a function of k. The amplitude is related to the phase shift experienced by the photoelectron as it encounters the potential of the back‐scattering atom. The other term within the sin function is the phase shift of the central atom, δ_c, which is experienced twice: when leaving and returning to the central atom. These atomic factors used to be estimated empirically, but now are generally calculated reliably. They can therefore be used to fingerprint the row of the periodic table of the back‐scattering atom.

The other terms in equation 2.6 generally restrict the range and precision of the information about the structure around the absorbing atom. The range is restricted as the photoelectron wave will become less dense with increased distance (by ). Also, the mean free path of the photoelectron, λ(k), will also ensure that the information observed in the EXAFS region is local to less than 1 nm. The precision of the coordination number, N_j, is compromised by two other amplitude related factors in equation 2.6. The factor allows for the fact that not all of the absorbed x‐rays lead to single electron process and therefore to EXAFS. In practice this value is about 0.8 ± 0.1, and this alone can result in a 10% margin of error in the coordination number. The last term, an exponential, contains the Debye‐Waller factor, σ (Å). Based upon there being a Gaussian distribution of interatomic distance within a shell, this estimates the spread of the distribution. This disorder can be either static, due to an intrinsic variation within the structure, or dynamic, due to the spread of distance caused by thermal vibrations. The timescale of the x‐ray transition is much faster than the timescale of a vibration and so XAFS probes the distributions of the interatomic distances in the vibrational envelope. This factor will reduce with lowered sample temperatures. The k dependence of this exponential means that an increased Debye‐Waller factor will dampen the EXAFS especially at higher k. Not only might that make it difficult to unravel estimate the coordination number, it will also reduce the accuracy of the inter‐atomic distance, R_j, by reducing the range of the useful data.

The effects of changes in R_j and the Debye‐Waller factor, can be envisaged using equation 2.7, an extract from equation 2.6.

(2.7)

In Figure 2.14a, the effect of changing the interatomic distance is shown. When the distance is increased the k interval for an oscillation is decreased. In different parts of the spectrum the two waves vary between being nearly completely out of phase to being in phase, and a beat will result. The plot shows the value of having a large k range to distinguish these two distances accurately. The effect of increasing the Debye‐Waller factor is shown in Figure 2.14b. The damping of the oscillation is increased and this is more serious at higher k values, effectively truncating the useable k range.

Figure 2.14 a) The oscillations calculated for an interatomic distance, R, of 1.8 (solid line) and 2.1 Å (dashed line), taking N as 1 and 2σ² as 0.005 Ų. b) Oscillations calculated for 2σ² as 0.005 (solid line) and 0.015 Ų (dashed line), taking N as 1 and R as 2.1 Å.

This word of warning should not be taken as a counsel of despair. EXAFS can provide significant structural information about the 5–6 Å space around an absorbing atom. As an example, the W L₃ EXAFS pattern of the cluster anion, [W₆O₁₉]²⁻, shows complicated patterns indicative of contributions from many shells (Figure 2.15a). As well as the strong features below k = 9 Å⁻¹ attributable in significant part to W‐O back‐scattering, there is clear evidence of another envelop peaking near 16 Å⁻¹. This is characteristic of back‐scattering from a high Z element, tungsten in this case. The amplitude of the Fourier transform of this pattern (Figure 2.15b) shows two strong components due to the terminal and edge‐bridging O atoms. The distances in the Fourier transform are shorter than the expected bond lengths due to the effect of the phase shift terms in the sin function in equation 2.6. There are strong contributions too from W‐O‐cis‐W back‐scattering, even though these are non‐bonded interactions. The highest R contributions apparent result from back‐scattering from the W‐O‐trans‐W unit, which is about 4.6 Å distant. These are relatively long distances to observe in the EXAFS of a molecular ion, especially with no metal‐metal bonding. This is due to a combination of the high overall symmetry, the high atomic number of the non‐bonded back‐scatterers, and the relatively low Debye‐Waller factors created by the cage structure.

Figure 2.15 a) The k³χ(k) of the W L₃ edge of (NBu₄)₂[W₆O₁₉] (inset) and b) the magnitude of the Fourier transform of this pattern.

(Source: Diamond, B18, data from Richard Ilsley)

2.2 Secondary Emissions

The excited state created by the absorption of the x‐ray has a short lifetime. Indeed, core‐hole lifetimes (τ) can afford very significant Heisenberg uncertainty broadening (Γ) limiting the resolution of XANES features (equation 2.8); this effect increases for deeper core holes. For example, the broadening at the sulfur K edge at 2.47 keV is rather modest at 0.59 eV, as it is for the Fe K edge (1.25 eV at 7.11 keV). However, for lead the line broadening becomes very large: 60 eV for the K edge at 88 keV, and would broaden EXAFS as well as XANES. This is one reason why the preferred edge for lead is the L₃ (13 keV) with a broadening of 5.8 eV. The decay of the core‐hole is accompanied by energy loss either by electron excitation or the emission of a fluorescent photon.

(2.8)

2.2.1 X‐Ray Fluorescence

As an example, the 1 s core hole created above the K absorption edge of an element can be filled with an electron dropping down from a higher core level. A dipole‐allowed transition, from a 2p or 3p dropping down to the 1 s core‐hole can result in the emission of x‐ray fluorescence (Figure 2.16). Both of these pathways occur and give rise to different x‐ray emissions with energies characteristic of the element. Different nomenclatures of x‐ray emissions exist. The labels start with the core hole to be filled. So the prominent emissions for copper that are emanate from anode sources in many x‐ray diffractometers, Kα₁ and Kα₂, are also labeled by the destination and source states, KL₃ and KL₂, respectively (Table 2.2).[2]

Figure 2.16 a) Creation of a core hole and b) its relaxation through x‐ray emission.

Table 2.2 X‐ray absorption edge energies and emission energies for copper.

Edge or emission	K	Kα1	Kα2	L3	L2
Transition label	K	KL3	KL2	L3	L2
Orbitals involved	1 s	1 s‐2p3/2	1 s‐2p1/2	2p3/2	2p1/2
Observed energies (eV)	8980.5	8047.8	8027.8	932.5	952.5
Estimated from absorption edges (eV)	–	8048	8028	–	–

The close agreement between the measured x‐ray emissions and those estimated from the absorption edge energies of the assigned transitions shows the validity of those descriptions.

X‐ray fluorescence provides an alternative method to transmission to measure XAFS spectra (Section 4.3.3), and this can be the method of choice for dilute or thin samples. Under those conditions, the fluorescence yield is linearly related to the x‐ray absorption. One of the key factors governing the sensitivity of the fluorescence yield sensitivity is the proportion of the core‐hole that is relaxed through fluorescence rather than electron emission. For the K edges, this increases with a sigmoid curve from very small values for light elements (~2% for carbon) to over 90% for high Z elements (95% for gold) with the crossover at Z = 30 (Zn). Emissions related to the L absorption edges also increase in probability with atomic number.

2.2.2 Electron Emission

The alternative method accommodating the energy released through the filling of a core‐hole by a higher energy electron is by electron emission (Figure 2.17). In this mechanism, the core‐hole is filled leaving a vacancy in a higher core level. The electron from a third core level is emitted, and this is referred to as an Auger process, which leaves two core‐holes. So, for example, at a K absorption edge, the K shell core‐hole created (1 s orbital) could be filled by an electron falling from the L₁ shell (2 s) and much of the energy released by the photo‐ejection of a 2p electron is to overcome the binding energy, with the remainder being kinetic energy. The unpaired electrons in the resulting core levels can couple to give particular Russell Saunders states. In this example a transition would be labeled KL₁L₂₃(¹P). The ion would have 2 s and 2p vacancies, which couple to form a singlet state. Auger transitions can also involve valence orbitals (generally labeled as V).

Figure 2.17 a) Creation of a core hole and b) its relaxation through Auger electron emission.

There are many other secondary processes that cause electron loss. The current from all of these processes can be used to monitor x‐ray absorption by electron yield. In this simplest case the total current is monitored as a drain current from the sample. This can include a significant background, but sampling can be refined with energy selection right down to a single Auger transition of the x‐ray absorbing element.

2.2.3 Resonant Inelastic X‐Ray Scattering or Spectroscopy (RIXS)

Resonant inelastic scattering is also known as Resonant x‐ray Raman. This process is shown in Figure 2.18. The incoming photon is of the appropriate energy to excite into the XANES region, either a low‐lying valence or continuum state. The outgoing fluorescent photon may result from relaxation from the valence electrons into the vacated core hole or by a dipole allowed core to core‐hole transition emitting a lower energy photon. The energy difference between incoming and outgoing photon energies is the inelastically transferred energy. So either the higher level core spectrum or even the valence band of a material can be probed with the energy source being that of the incoming x‐ray, similar to vibrational spectra being probed by inelastic scattering of visible light in Raman spectroscopy.

Figure 2.18 A x‐ray resonant inelastic scattering process or Resonant x‐ray Raman. a) L₃ edge absorption and b) L₃M₁ emission.

2.3 Effects of Polarization

With laboratory x‐ray sources, the light has no polarization properties. Light from a synchrotron source is intrinsically polarized in the horizontal plane. Depending upon the viewing point, or the particular magnetic array type providing the light, polarization can be plane or circular.

2.3.1 Plane (Linear) Polarization

Many materials are isotropic in nature, being presented either as a solution or as a randomly oriented powder. For these cases, and for cubic crystals, the polarization in the light source is completely nullified. However, most crystalline samples or a surface will engender some anisotropy and the orientation of the polarization relative to the sample will become important. Then the orientation of a virtual state in valence band may cause the absorption to be reduced or lost. As in some NMR experiments, the isotropic spectrum can be reclaimed by magic‐angle spinning with the sample oriented at ~54.7° (of cosine 1/√3). In principle thorough polarization‐dependence studies can provide orientational information, for example, of how molecules are adsorbed on a surface.

2.3.2 Circular Polarization

Circularly polarized x‐radiation can also be sampled from storage ring light sources. It may be thought that this could be utilized to study optically active (chiral) materials in solution. However, there is no such effect observable in isotropic media. In crystalline samples, or in solution oriented in liquid crystal solvents, pre‐edge features have been shown to display circular dichroism.

2.3.3 Magnetic Dichroism

The more prevalent application of circular polarized x‐rays in XAFS spectroscopy is to investigate x‐ray magnetic circular dichroism (XMCD). In ferromagnetic materials, it has been shown that the absorption at the L₂ and L₃ edges display XMCD effects in magnetic fields. Antiferromagnetic, ferromagnetic and ferromagnetic materials display magnetic dichroism with linear polarization (XMLD). Magnetic dichroism arises from coupling between the between the 2p⁵ electrons with the valence electrons in the excited states (3dⁿ⁺¹). For example, for Ni(II) (3d⁸), the excited state would be 2p⁵3d⁹. Right‐ and left‐hand polarized light cause changes of +1 and ‐1 to the magnetic J quantum number, and it is by virtue of the J states that the L₃ and L₂ edges differ. An example of this effect is shown in Figure 2.19, recorded in total electron yield from a thin film sample.

Figure 2.19 Example of the XMCD effect (dotted) at the L₃ and L₂ edges of an iron oxide. XAFS with right (solid) and left (dashed) polarized light. Recorded by total electron yield on a thin film sample.

(Source: Diamond, I06, data from Sarnjeet Dhesi)

2.4 Questions

1. 1 The energies and linewidths of absorption edges of molybdenum are given in Table 2.3.
   1. Estimate the energies of the following emission lines: Kα₁, Kα₂, Kβ₁, Kβ₂.
   2. From the data in the table, assess which edges may provide most information in the XANES and EXAFS region. What would be the observable k range for the L₃ edge (see equation 2.4)?
2. 2 The K absorption edge of sulfur occurs at 2472 eV and the M₁ to M₅ edges of lead at 3850.7, 3554.2, 3066.4, 2585.6, and 2484.0, respectively. For the compound PbMoS₄, which edge(s) could be studied without complication from the other elements, and over what energy range? Which edge might be the best to pick to provide some EXAFS data to k = 8 Å⁻¹?
3. 3 The structure of Na₂MoO₄ contains tetrahedral anions, with a Mo‐O distance of 1.97 Å. Using equation 2.6, how many complete oscillations should be observed if a k range of 12 Å⁻¹ were recorded?
4. 4 The related salt Na₂Mo₂O₇ has a chain structure with two Mo sites in the crystal: one tetrahedral and the other octahedral.
   1. How might these be distinguished in the XANES features of the K and L₃ edges?
   2. The Mo‐O distances in the four‐coordinate tetrahedral site have an average value of 1.756 Å with a mean square deviation of 0.004 Ų. The six‐coordinate octahedral site has a larger average (1.95 Å) and spread (0.227 Ų).
   3. Use equation 2.7 for these two sites, taking the static disorder as σ² and plot χ up to k = 15 Å⁻¹. Note the difference in the observable k range in these two cases.
   4. The octahedral site is made up of three pairs of Mo‐O bond lengths of 1.685, 1.899, and 2.267 Å, which may be considered as three different shells. Remodel the χ function as the sum of these three shells each with a coordination number of 2, taking 2σ² as 0.005 Ų. Note the beats between the shells and that the simple one‐shell model of the octahedral site misses out much detail.

Table 2.3 The energies and line widths (eV) of the K, L, and M absorption edges of molybdenum.

Edge	K	L1	L2	L3	M1	M2	M3	M4	M5
Energy	20000.4	2865.5	2625.1	2520.2	506.3	411.6	394.0	231.1	227.9
Width	4.5	4.25	1.97	1.78	~6	2.2	2.2	0.16	0.14

References

1. 1 ‘Tables of x‐ray mass attenuation coefficients and mass energy‐absorption coefficients from 1 keV to 20 MeV for elements Z = 1 to 92…’, J. H. Hubbell, S. M. Selzer, www.nist.gov/pml/data/xraycoef/index.cfm.
2. 2 ‘X‐ray Wavelengths’, J. A. Bearden, Rev. Modern Phys., 1967, 39, 78–124.
3. 3 ‘X‐ray Data Booklet’, Ed. A. C. Thompson, Center for X‐ray Optics and Advanced Light Source, LNBL/PUB‐490 Rev. 3, 2009. (http://adb.lbl.gov/xdb‐new.pdf). On‐line calculations from http://henke.lbl.gov/optical_constants.
4. 4 ‘Cr K‐edge XANES: ligand and oxidation state dependence. What is oxidation state?’, M Tromp, J. Moulin, G. Reid, J. Evans, AIP Conf. Proc., 2007, 282, 699–701.