1 Chapter 1 X-ray Absorption Fine Structure (EXAFS) 1.1 What is EXAFS? X-ray absorption fine structure (EXAFS, XAFS) is an oscillatory modulation in the X-ray absorption coefficient on the high-energy side of an absorption edge (Lee et al., 1981) Analysis of EXAFS can yield the numbers and types of atoms in the immediate environment of the absorbing atom, and accurate absorber-neighbour distances. Figure 1.1 shows the K X-ray absorption edge of CuO which arises from the absorption of a photon of energy above 9000 ev, promoting a 1s core electron to the continuum. 0.8 0.7 0.6 µ t (cm-1) 0.5 0.4 0.3 0.2 8400 8600 8800 9000 9200 9400 9600 9800 10000 Energy (ev) Figure 1.1 The K X-ray absorption edge of CuO. This data was recorded by Dr. G. A. Williams (Australian Radiation Laboratory) and is used in all the illustrations in this chapter. The EXAFS is defined: χ= µ µ µ 0 0 (1.1) where µ 0 is the hypothetical smooth background absorption coefficient due to the transition of interest; and µ is the observed absorbance.
2 Figure 1.2 shows the EXAFS from CuO. 0.20 0.15 0.10 χ 0.05 0.00-0.05-0.10-0.15 8900 9000 9100 9200 9300 9400 9500 9600 9700 9800 9900 10000 10100 Energy (ev) Figure 1.2 K-edge EXAFS from CuO. 1.2 The origin of the EXAFS oscillations The EXAFS arises from the interaction of the absorbing atom with photoelectron waves backscattered by neighbouring atoms. If the binding energy of the core electron is E 0, absorption of an X-ray photon of energy E > E 0 generates a photoelectron of energy E - E 0. According to the de Broglie relation this photoelectron has wavelength: λ e = h / p e (1.2) where h is Planck's constant and p e is the momentum of the photoelectron. The energy of the photoelectron is related to the momentum by the kinetic energy equation: E = p 2 / 2m (1.3) e e e where m e is the mass of the electron.
3 Combining equations (1.2) and (1.3), the wavelength of the photoelectron is: λ e e e e = h/ 2m E = h/ 2m ( E E0 ) (1.4) Figure 1.3 shows an absorbing atom emitting a photoelectron wave of wavelength λ e. This wave is scattered from another atom at a distance R as and returns to the absorber. Emitted wave Scattered wave Absorber A S Scatterer e R as Figure 1.3 A photoelectron wave emitted from the absorbing atom (A) is back-scattered by a scattering atom (S). The back-scattered wave modifies the final-state wavefunction at the absorber. If the emitted and back-scattered waves are in phase, the wavefunction is increased. If the emitted and back-scattered waves are out of phase, the wavefunction is decreased. To complete the round trip, the photoelectron wave travels a distance 2R as. This corresponds to 2R as / λ e wavelengths or a change in phase of: phase = 2π( 2Ras / λe ) (1.5) In EXAFS analysis, the quantity 2π / λ e is called the photoelectron wave vector, k: 1 k = 2π/ λ = h 2m ( E E ) (1.6) e e 0 where h is h /2π.
4 The phase difference at the absorbing atom between the emitted and scattered waves is 2kR as. As the phase of the wave is also changed in traversing the potentials of the absorbing and scattering atoms, a correction α as (k) must be added: phase = kr + α ( k) (1.7) 2 as as The absorbance is increased when the waves are in phase and decreased when the waves are out of phase. The dipole-coupled absorption cross-section is: * σ = ê ψ i rˆ ψ f dτ 2 (1.8) where ê is the electric-field polarisation vector of the X-ray photon, ψ i is the initial-state wavefunction, which has significant magnitude only near the absorbing atom, and ψ f is the final-state wavefunction. The absorbance varies with the magnitude of ψ f in the vicinity of the absorber. This is increased by constructive interference when the waves are in phase, and decreased by destructive interference when the waves are out of phase. 1.3 Multiple-scattering The preceding discussion assumed that the photoelectron wave is scattered from only one other atom before returning to the absorber. This is known as single-scattering. In practice, the wave may be scattered successively from a number of atoms. This is known as multiple-scattering. In this case, R as is half the total distance travelled by the wave. For a more detailed description of the calculation of EXAFS, see chapter 2.
5 1.4 Data collection 1.4.1 Generating monochromatic X-rays The vast majority of X-ray absorption spectroscopic (XAS) measurements rely on the continuous spectrum of X-rays produced by synchrotrons (Gurman, 1995). This radiation is emitted when electrons or positrons, circulating in a ring at near the speed of light, are accelerated by a magnetic field. The X-rays are emitted in the direction of the electron or positron beam and are polarised in the plane of the ring (figure 1.4). ê monochromator ê e- or e+ beam magnet monochromatic X-ray beam θ "white" X-ray beam Figure 1.4 Generation of the monochromatic X-ray beam. The "white" X-ray beam is generated from a synchrotron and passed through a monochromator consisting of 2 crystals, usually of silicon. Radiation satisfying the Bragg condition nλ=2d sinθ is reflected, where d is the lattice spacing of the crystal plane. The desired X-ray energy is selected by varying θ. The X-rays are polarised in the plane of the ring. ê is the electric-field polarisation vector. Only 1 magnet of the many placed around the ring is shown. The desired energy is selected from the continuous spectrum using a monochromator. An X-ray monochromator is typically constructed of two crystals. The crystals are cut parallel to a lattice plane and arranged as shown in figure 1.4. Only the X-rays satisfying the Bragg condition: nλ =2d sinθ (n = 1, 2,...) (1.9) where d is the lattice spacing, are reflected. Although the monochromator will transmit undesired higher harmonics (n > 1), these can be removed in a number of ways. Two common methods are to use a mirror or to detune the monochromator. The first method uses a mirror reflecting X-rays only below a
6 certain energy (Lee et al., 1981). The second method rejects the harmonics by rotating the second monochromator crystal slightly out of parallel to the first. This takes advantage of the fact that the rocking curves for the higher harmonics are much narrower and are displaced with respect to the fundamental (Hasnain, 1987). 1.4.2 Transmission XAS measurements The simplest type of XAS measurement is transmission XAS. In a transmission experiment, the intensity of the X-ray beam is measured before and after a sample and the absorbance (µ t ) calculated using the expression: µ x = ln t ( I I ) 1 / 0 (1.10) where x is the thickness of the sample. x is usually ignored as it is eliminated when the EXAFS is extracted. The intensity of the X-ray beam is typically measured using ionisation detectors. These detectors consist of a chamber filled with a gas or gas mixture. The X-ray beam passes through windows at each end of the chamber between two oppositely-charged plates. The (photoion) current passing between the plates is proportional to the X-ray intensity (Lee et al., 1981). Figure 1.5 represents the detection apparatus for a transmission EXAFS experiment. As shown in this figure, a third detector is often added to allow the concurrent measurement of the XAS from a standard. This standard spectrum is used to calibrate the energy scale using the position of features (transitions and inflection points) with known energy. I 2 I 1 I 0 standard sample beam Figure 1.5 Typical apparatus for a transmission XAS experiment. I 0, I 1 and I 2 are ionisation detectors. A minimal experiment requires only I 0 and I 1. The addition of I 2 permits the XAS of a standard such as a metal foil to be measured. This standard XAS is used for accurate energy calibration.
7 1.4.3 Fluorescence XAS measurements Fluorescence experiments are much more sensitive than transmission and are typically used with samples in which the absorbing atom is dilute (Lee et al., 1981). In a fluorescence experiment, the absorbance of the sample is measured by monitoring the intensity of the X-ray fluorescence produced when higher-shell electrons relax into the hole left by the photoelectron. If F is the intensity of the fluorescence X-rays, the absorption coefficient is: µ x = CF ( / I0 ) (1.11) where C is approximately constant. C is normally neglected as it is eliminated when the EXAFS is extracted. Figure 1.6 represents the detection apparatus for a fluorescence EXAFS experiment. The sample is commonly oriented at 45 to the beam and the fluorescence detector placed at 90. Because the X-rays do not have to pass through the fluorescence detector, a solidstate detector or an ionisation detector filled with a gas of high X-ray cross-section to maximise detection is normally used. fluorescence detector I 2 I 1 I 0 standard sample beam Figure 1.6 Typical apparatus for a fluorescence XAS experiment. The fluorescence detector is usually an ionisation detector or a solid-state detector. A minimal experiment requires only I 0 and the fluorescence detector.
8 1.4.4 Polarised XAS measurements The XAS collected from solutions and powders is unpolarised. Polarised XAS can be collected from anisotropic samples such as crystals. To collect polarised XAS, the sample is placed in a known orientation with respect to the (horizontal) electric-field polarisation vector of the X-ray beam. In the present work, this was accomplished by identifying the crystal axes using X-ray precession photographs and placing the crystal in the desired orientation using the arcs on a goniometer head and a crystal orienter (figure 1.7). laser fluorescence detector beam I 2 standard I 1 I 0 φ ê crystal orienter χ crystal inside capillary (magnified) Figure 1.7 Typical apparatus used in collecting polarised XAS data from oriented single crystals. The laser can be swapped with the standard and I 2 and is used to centre the crystal in the beam path. The crystal is placed in the desired orientation with respect to the (horizontal) polarisation vector (ê) using the φ and χ arcs of the crystal orienter. 1.4.5 Low-temperature XAS measurements Any of the experiments discussed above may be conducted at low temperatures. For this purpose, the sample is enclosed in a liquid-n 2 or liquid-he cryostat. Windows in the cryostat sample chamber permit the passage of X-rays.
9 1.5 Data reduction (extracting the EXAFS) Data reduction refers to the extraction of the EXAFS curve from the raw absorption data. To extract the EXAFS, two contributions to the total absorbance must be removed: 1. The underlying background absorbance (the absorbance that would be observed in the absence of an edge). 2. The featureless background edge absorbance. The following discussion refers to the method used in the current work and implemented in the program SPLINE (Ellis, 1995a). 1.5.1 Removing the underlying background absorbance The background or "pre-edge" absorbance is estimated by fitting a polynomial function to the absorbance curve prior to the edge using a least-squares procedure and extrapolating to the end of the data. In the example of figure 1.8, the pre-edge curve was obtained by fitting a quadratic polynomial to the CuO absorbance between the energies 8479 ev and 8928 ev. 0.8 0.7 0.6 µ (cm-1) 0.5 0.4 0.3 0.2 0.1 Fitted region 8400 8600 8800 9000 9200 9400 9600 9800 10000 Energy (ev) Figure 1.8 The K X-ray absorption edge of CuO and the background absorbance estimated by fitting a quadratic curve to the absorbance between 8479 ev and 8928 ev.
10 1.5.2 Normalisation After subtraction of the pre-edge background, the absorbance is scaled to an edge step of 1.0 so that the final EXAFS is relative to the edge (µ 0 ) as required by equation (1.1). 1.5.3 Removing the smooth edge background The hypothetical smooth background absorbance above the edge is estimated by fitting a polynomial spline function to the normalised absorbance. The polynomial spline curve consists of one or more polynomial segments. Adjacent segments are constrained to meet and to have the same first derivative, ensuring that the junctions are smooth. A leastsquares procedure is used to fit the spline curve to the normalised absorbance. The weight given to the data points is increased with increasing energy as the fit has to be better at high energy due to the decreasing magnitude of the EXAFS oscillations. The curve in figure 1.9 was obtained by fitting a 3-segment polynomial spline to the normalised CuO absorbance. The polynomial coefficients were calculated to minimise Σ [k 3 (µ normalised (E)-µ spline (E))] 2 for all points above E 0, set arbitrarily to 9000 ev. 1.2 1.0 µ (cm-1) 0.8 0.6 0.4 Order 2 Order 3 Order 3 0.2 0.0 8400 8600 8800 9000 9200 9400 9600 9800 10000 Energy (ev) Figure 1.9 Normalised K X-ray absorption edge of CuO and the featureless background edge absorbance estimated using a polynomial spline curve. The spline curve consisted of a segment of order 2 from 9025 ev to 9240 ev, and segments of order 3 from 9240 ev to 9630 ev and 9630 ev to 10015 ev.
11 1.5.4 Compensating for decreasing µ 0 After subtraction of the spline curve, the curve in figure 1.10 was obtained. 0.20 0.15 0.10 µ µ spline (cm-1) 0.05 0.00-0.05-0.10-0.15 8900 9000 9100 9200 9300 9400 9500 9600 9700 9800 9900 10000 10100 Energy (ev) Figure 1.10 Difference between the normalised K-edge absorption from CuO and the featureless background edge absorbance estimated using a polynomial spline curve. To obtain the EXAFS, the difference between the normalised absorbance and the spline curve is divided by the normalised background absorbance due to the edge (µ 0,normalised ). µ 0,normalised is estimated using the expression: µ 4 ( C C ) ( D D ) 3 0, normalised = λ a b λ a b (1.12) where λ is the X-ray wavelength and C b and D b and C a and D a are the Victoreen coefficients before and after the edge as tabulated in the International Tables for X-ray Crystallography (Macgillavry & Rieck, 1962). The estimated normalised edge absorbance is compared to the normalised K-edge absorbance of CuO in figure 1.11.
12 1.2 1.0 µ (cm-1) 0.8 0.6 0.4 0.2 0.0 8400 8600 8800 9000 9200 9400 9600 9800 10000 Energy (ev) Figure 1.11 Normalised absorbance calculated using expression (1.13) compared to the normalised K-edge absorbance of CuO. The final EXAFS, plotted as a function of k and multiplied by k 3 to compensate for the rapid attenuation with increasing energy, is shown in figure 1.12. 6 4 2 k 3 χ 0-2 -4-6 0 2 4 6 8 10 12 14 16-1 k (Å ) Figure 1.12 K-edge EXAFS from CuO. The EXAFS has been graphed as a function of k and multiplied by k 3 to compensate for the rapid attenuation with increasing energy. k was calculated using E 0 = 9000 ev.