X-rays

This chapter provides a brief description of the basic X-ray physics needed to design XAFS experiments. We start with the basics.

X-rays are short-wavelength electromagnetic (EM) radiation; except for their wavelength, they are essentially the same as radio waves, microwaves, infrared, visible, ultraviolet, and gamma radiation. The frequency f is related to the wavelength λ by fλ = c, where c is the speed of light, ≈ 3 × 108 m/s.

In free space, EM waves are transverse: the electric and magnetic field vectors of the wave are perpendicular to each other, and also to the direction of propagation. The electric and magnetic field vectors oscillate in phase, and their magnitudes are proportional to each other. The direction of the electric field is described by the “electric polarization vector”, which is a unit vector in the direction of the wave's electric field vector. The direction of wave propagation is given by the wave vector where.

From a quantum perspective, the electromagnetic waves of classical physics consist of swarms of photons, which carry energy, linear momentum, and angular momentum. Such a wave is illustrated in Figure 2.1. The wavelength λ of all particles, including photons and electrons, is related to their momentum p through the De Broglie relation λ = h/p, where h is Planck's constant. Similarly, the particle frequency f is related to the energy E by f = E/h.

Requirements for XAFS experiments

The features of interest in XAFS spectra consist of small variations in the absorption coefficient µ(E), which can be determined directly in a transmission experiment, or indirectly by measuring the variation in the intensity of specific fluorescence emission lines as the energy of the incident beam is scanned over an absorption edge. Sometimes useful information, such as edge shifts, can be obtained from XANES spectra that are noisy, but in general very good signal to noise ratio (S/N) is required for EXAFS analysis and detailed XANES analysis. For EXAFS measurements one requires S/N ratios better than 103 in order to determine the spectra accurately enough in the region ≈ 600–1000 eV above the absorption edge. It is generally necessary to measure spectra at least this far above the edge in order to get adequate spatial resolution. The EXAFS signal may damp out quickly above the edge because of disorder and/or low-Z scatterers, rapidly plunging the signal into the noise, unless the noise level is kept very low.

An intense beam is required to obtain good data in a reasonable time frame (minutes to hours): on the order 1010 photons/sec or better within the required energy bandwidth of an eV or so. For this reason, synchrotron radiation sources are by far the preferred sources of X-rays for XAFS experiments.

Introduction

XAFS theory has advanced significantly over the last several decades, driven by creative theorists, experimental demand, and Moore's law. It continues to evolve at a rapid rate. XAFS theory has benefited from and contributed to advances in many-body physics, quantum field theory, and scattering theory. Some current areas of focus include work to improve the accuracy of computing non-spherical molecular potentials, vibrational effects, and multilelectron excitations. Efforts to improve parallelization of codes to take better advantage of multiple processor cores, multiple processors, graphics processing units, computational grids, and cloud computing are also under development.

Theoretical advances have transformed the practice of data interpretation and analysis. Computer programs are now readily available for the calculation of X-ray Absorption spectra with very good accuracy in the EXAFS region, and useful accuracy in the XANES. A number of complementary theoretical approaches and the computer programs that implement them are widely disseminated and used, among them EXCURV, FEFF, GNXAS, MXAN, and FDMNES.

There is little fundamental disagreement about the correct basic physics underlying all these approaches. They differ principally in the sets of approximations that are used to make possible efficient calculation of the spectra, and in the computational algorithms that are employed. Recent progress has shown that one theoretical approach, specifically the Real Space Multiple Scattering (RSMS) formalism, is useful also for calculating X-ray emission, X-ray magnetic circular dichroism, X-ray elastic and inelastic scattering/X-ray Raman Scattering, and dielectric response functions.

In the process of data analysis one creates a hypothetical model of the data containing unknown parameters or functions that are to be determined through comparison with experimental data. Once the model is established, the goal is to identify and then describe the regions within the space of parameters that are consistent with the data, within the experimental and theoretical uncertainties.

The “forward” problem – calculating spectra for a hypothetical set of structural parameters – has been the principal activity in XAFS theory in recent decades. The inverse problem – determining structural parameters from a set of data – usually is handled through fitting and some kind of statistical analysis to generate confidence limits on the determined parameters. Nevertheless a variety of methods have been developed for solving the inverse problem in a direct manner. Some of these methods are briefly described at the end of this chapter.

XAFS data depend on the types and dispositions of atoms and the variation of those structural parameters over the sample volume. Modeling the structural variation in a sufficiently accurate manner without introducing too many parameters is often the trickiest part of formulating a fitting model. Near-edge spectra furthermore can be sensitive to oxidation states, either directly by introducing vacancies in bound state orbitals or affecting the charge distribution, or indirectly by inducing structural changes.

As described in Chapter 4, EXAFS has a standard parameterization in terms of multiple scattering paths that is descended from the original formulation by Stern, Sayers, and Lytle.

There is perennial interest in reference XAFS spectra. This appendix contains a small sample of µ data taken by the author and collaborators. Preedge fits are removed to improve visibility, but that is all.

The first three spectra are from study of multiple scattering in gases. A Si(111) monochromator was used on a bend magnet beamline (X9) at NSLS. This study permitted isolation of the Cl multiple scattering signal in molecular gases simply by taking linear combinations of raw spectra. Weak backscattering from hydrogen was also noted. The triangle path scattering is evident right above the “white line”; similar features are seen in ZnS4 tetrahedral sites such as are present in ZnS and various enzymes, such as Aspartate Transcarbamoylase. The GeCl4 spectrum contains single scattering plus multiple scattering among the Cl; GeH3Cl contains mostly single scattering from the Cl and weak scattering from H, (the weak multiple scattering between the central atom and the Cl and multiple scattering among the hydrogens are neglected). GeH4 principally has only weak backscattering from the hydrogens.

The other spectra were taken with a Si(220) monochromator and an 18 KG wiggler at SSRL. The next three spectra are KMnO4 at 80K, 160K, 300K sample temperature. Note the temperature dependence of the high frequency structure at the edge. This is due to single scattering contributions from non-nearest neighbor atoms, which are loosely bound, and therefore have large temperature dependent σ2.

What is XAFS?

X-ray Absorption Fine Structure (XAFS) spectroscopy is a unique tool for studying, at the atomic and molecular scale, the local structure around selected elements that are contained within a material. XAFS can be applied not only to crystals, but also to materials that possess little or no long-range translational order: amorphous systems, glasses, quasicrystals, disordered films, membranes, solutions, liquids, metalloproteins – even molecular gases. This versatility allows it to be used in a wide variety of disciplines: physics, chemistry, biology, biophysics, medicine, engineering, environmental science, materials science, and geology.

The basic physical quantity that is measured in XAFS is the X-ray absorption coefficient µ(E), which describes how strongly X-rays are absorbed as a function of X-ray energy E. Generally µ(E) smoothly decreases as the energy increases (approximately as 1/E3), i.e. the X-rays become more penetrating. However, at specific energies that are characteristic of the atoms in the material, there are sudden increases called X-ray absorption edges. These occur when the X-ray photon has sufficient energy to liberate electrons from the low-energy bound states in the atoms. The cross section, a quantity that is proportional to µ(E), is shown in Figure 1.1 for the element platinum. Experimental data for MnO and KMnO4 are shown in Figures 1.2 and 1.3.

Absorption edges were first measured in 1913 by Maurice De Broglie, the older brother of quantum mechanics pioneer Louis De Broglie.

Despite advances in energy dispersive detector technology, X-ray filters remain a useful tool for optimizing fluorescence XAFS experiments. They are essential when using fluorescence ionization chambers if diffractive analyzers are available.

The filter quality is an important but frequently ignored parameter that can determine the success or failure of a fluorescence XAFS experiment. In this appendix we define filter quality, describe its effects, and derive equations and rules of thumb for choosing the optimal filter thickness during an experiment. We will see that the optimal thickness depends on the background to signal ratio, the quality Q of available filters, and the effectiveness of the slits.

This appendix is concerned with minimizing high-energy (usually elastically scattered) background. Filters are used to preferentially absorb the X-rays that are scattered from dilute samples. Statistical fluctuations in the number of scattered photons are a principal source of noise in fluorescence EXAFS experiments, and it is therefore desirable to minimize the scattered background as much as possible, without attenuating the signal significantly. As described above, normally the main constituent of a filter is an element which has an absorption edge that falls between the strongest fluorescence lines and the absorption edge of the element of interest (see Figure 3.12). In many cases a suitable filter can be constructed using the element of atomic number (Z) one less than that in the sample (a “Z – 1 filter”).