Experimental and Theoretical Developments in Extended X-Ray Absorption Fine Structure (EXAFS) Spectroscopy

This thesis presents a detailed study of the physical basis of Extended X-ray Absorption Fine Structure Spectroscopy (EXAFS). A single EXAFS scattering formalism is presented that allows a rigorous treatment of the central atom potential. A study of the physical basis for determining structural information from EXAFS data is also presented.

In addition, a scheme is introduced to determine the nature of the scattering atom in EXAFS experiments.

CHAPTER VI Figure 1

CHAPTER VII Figure 1

CHAPTER I

1.1.1) is a valid description of EXAFS, then it should be possible to reverse this expression to obtain the distances rj. These peaks do not appear at the right shell distances due to the presence of the phase function 9j(k) in the argument of each sine wave. Increased understanding of the physical basis of the EXAFS effect has been paralleled by the rapid development of instruments used to measure EXAFS spectra.

In VIII. chapter presents a new scheme for determining the nature of a scattered atom.

CHAPTER II

The second term in Eq. 2.4.13) is canceled by the average of the single scattering junction originating from the atoms i and j. The simplicity of our approach lies in the expansion of the scattering amplitudes of neighboring atoms around the origin. In the alternative form of the Lippmann-Schwinger equation [i.e. the use of in Eq.

If such considerations are not made, serious errors can be made in the analysis of the data. Using the definite integral, . the radial integration in Eq. A4) can be performed to obtain. Diagrammatic representation of the first and second order terms in the expansion of the full T-operator in Eq.

CHAPTER III

A comprehensive treatment of the theory and application of final state interactions is given by Gillespie.l. The final state, , is the distorted wave e. photoelectron in the final state potential field. 3.3.6). define 1)/L(k,r) as the physically meaningful solution of the radial equation where fL(k) is the partial wave amplitude and ht(k) is a spherical Hankel function of the first kind.

An explicit expression for the Jost function can be obtained from Eqs. 3.3.11) It is clear that as the final state potential, Vf(r), becomes vanishingly small, FL(k) approaches unity and the amplitude in Eq. 3.3.10) reduces to the undisturbed system. Since the absorption coefficient is given by the square of the modulus of Eq. 3.3.10), it remains only to calculate the Jost function F1 (k) for the appropriate final state potential. 3.4.1) has been written as [lji !

The operator T can now be expanded in terms of individual operators associated with different scattering atoms in the final state as described in the previous chapter [see Eq. Under more restrictive assumptions about the asymptotic behavior of the final state potential at large distances, FL(k) can be shown to be an integral function of k. If the final state potential can be accurately modeled, the inelastic damping of the photoelectron wave can be calculated.

The origin of the EXAFS phenomenon can be easily explained within the framework of the formalism. However, in the case of EXAFS, the final state potential is centered on each of the neighboring atoms.

CHAPTER IV

The purpose of this paper is to provide a detailed study of the nature of the Debye-Waller factor in EXAFS. Previous studies of the Debye-Waller factor in EXAFS were exclusively concerned with single scattering events. This present work will discuss the nature of the Debye-Waller factor in EXAFS spectra that contain a significant multiple scattering component.

The exponential terms in Eq. 4.3.8) corresponds to the Debye-Waller factors for each of the scattering processes. The s 1 mode is A schematic representation of the normal modes of the three-atom system is shown in Fig. The temperature dependence of the Debye-Waller factor for the BeBr 2 system with three different bridging angles is shown in Fig.

We have provided a general description of the nature and origin of the Debye-Waller factor. The relative magnitudes of the Debye-Waller factors associated with the different scattering paths deserve comment. The temperature dependence of the Debye-Waller factors for the BeBr 2 system is shown in Fig.

This finding is consistent with the aforementioned dependence of the Debye-Waller triple scattering factor on the A1 stretching mode. In conclusion, the Debye-Waller factors in EXAFS spectra, which contain a significant multiple scattering component, are sensitive to the geometry of the system.

Fig. 2 (a) and 2 (b)). X~ (k,~) may be expressed as:

BRIDGING ANGLE 8 (DEGREES)

BEND

BRIDGING ANGLE 8

DEGREES)

TEMPERATURE (°K)

The scattering angle a can be written in terms of the unit vectors of the system. The second calculation requires a determination of the contribution of the scattering amplitude to the amplitude and phase of the modification factors. The modulus and phase of the scattering amplitudes together with their derivatives are shown in Fig.

The triple scattering path is the least sensitive, since a change in the bridge angle does not appreciably change the scattering path length. The temperature dependence of the double and triple scattering Debye–Waller factors is shown in Fig. The magnitude of the hyperbolic sine terms is inversely proportional to the bond distance [see Eq.

The temperature dependence of the hyperbolic sine term of double and triple scattering together with the angular damping factor is shown in the figure. Note that the magnitude of the triple-scattering hyperbolic sine term decreases with increasing temperature. The modulus and phase of the scattering amplitude for oxygen as a function of the scattering angle and k are shown in Fig.

However, since the phase derivative is positive, it is a type II double scattering modification. The triple scattering modification factors increase as the bridging angle increases due to the presence. It is the degree of correlation that determines the contribution of the modifying factors to the observed EXAFS.

The equilibrium positions of the central atom (o), the first nearest atom (j) and the second nearest atom {i) are shown as filled circles.

Table I. Normal frequencies of vibration for t~~

1 STRETCH

Schematic representation of the vectors that determine the change in the scattering angle a at atom j due to the displacement of all atoms in a certain normal way [see. Calculated frequencies of normal vibration modes for the Br 2o system as a function of the br-idging angle 9. -Waller factors for the three-atom system. a) Debye-Waller factor as a function of the bridge angle at 10°K.

The argument for the hyperbolic sine terms for the double and triple scattering trajectories, The angular damping factor is the exponent of the exponential term, which represents the attenuation of the EXAFS due to a change in the scattering angle 1. Note that the sign of the double scattering hyperbolic term is negative. a) These expressions as a function of bridge angle at l0°K. b) as a function of temperature at a bridge angle of 120°K. As a function of k for different scattering angles a. c) The rate of change of the modulus as a function of the scattering angle for different k values.

Phase of the scattering amplitude for oxygen (a) as a function of k for different scattering angles a. b) The rate of change of the phase as a function of the scattering angle for different k values. The sign of the modified triple scattering type I and II terms is positive, while all other terms are negative. Note that the modified type I terms must be multiplied by a cosine of the relevant argument, while the type II terms are multiplied by a sine.

SECOND SHELL SINGLE SCATTERING

To overcome this problem, helium beamlines are used along the x-ray path. To scan the x-ray wavelength, \ the source, monochromator and slits must be moved along the circumference of the Rowland circle. The observed intensity is then extrapolated through all the characteristic lines in the spectrum.

Due to the high temperature of the filament, the atoms from the filament diffuse to the anode. As a result, characteristic lines of filament material appear at the output of the X-ray source over a period of time. The X-ray monochromator used in the spectrometer is of the Johansson type shown in fig.

The crystal is then bent to the diameter of the circle so that the smooth surface extends to the circumference. Therefore, every point in the crystal diffracts x-rays at the same wavelength, which is determined by Eq. Note that bending moments are applied at both ends of the crystal with cylindrical pairs.

This allows translation of one or both ends of the crystal beyond bending. The angular range over which a crystal can diffract is called the acceptance angle ws of the crystal.

CALTECH EXAFS SPECTROMETER

JOHANSSON ARRANGEMENT

The output voltage is then Q/C, where Q is the total charge collected during the gasification process. EXAFS has been shown to be sensitive to the local environment of the absorbing atom.1 An expression for single scattering EXAFS can be written as: 2. This contribution can be calculated by fitting the pre-edge data to the so-called Victoreen formula, 11 v =aA3-bA4, where a and b are constants and A is the X-ray wavelength.

A data set of length equal to the length of the experiment is generated using the calculated parameters a and b above and then subtracted from the experimental spectrum to obtain the u c' corrected absorption coefficient. It is obvious that an accurate estimate of u0 is essential to compare the theoretical expression for EXAFS with experimental data. Since there is no analytical expression for u 0 (k) that is suitable for all systems, the researcher must estimate the points representing the background absorption.

These points, in most cases, .. are then subjected to a cubic fit to produce a data set of equal length to that of the experimental spectrum. 3 Recently, Cook and Sayers 4 presented an empirical set of criteria for background removal using the cubic spline method. However, the above methods are very flexible and ultimately depend on the judgment of the investigator.

Since E0 is a non-linear function of k, the value of E0 determines the frequency of the data in k space. Provided the connections are not very different, this method works quite well.s An alternative approach was to vary E0 until the peaks in the real and imaginary parts of the Fourier transform coincide.6 However, this latter method requires a knowledge of also many parameters to be useful in a study of unknown compounds.