Introduction to XAFS

Grant Bunker
Associate Professor, Physics
Illinois Institute of Technology

Outline

Overview of Tutorial
1: Overview of XAFS
2: Basic Experimental design and methods
3: Basic Theory
4: Basic Data Analysis
5: Intermediate Experimental methods
6: Intermediate Theory
7: Intermediate Data Analysis
8: Summary and new developments

1:  Overview of XAFS

What is XAFS?

X-Ray Absorption Fine Structure (XAFS) refers to modulations in x-ray absorption coefficient around an x-ray absorption edge. XAFS is often divided (somewhat arbitrarily) into "EXAFS" (Extended X-ray Absorption Fine Structure) and "XANES" (X-ray Absorption Near Edge Structure). 
The physical origin of EXAFS and XANES is basically the same, but several simplifying approximations are applicable in the EXAFS range, which permits a simpler quantitative analysis.  XANES and EXAFS provide complementary information.

XAFS of ZnS (Sphalerite)

A little history

XAFS was observed early in this Century by R. de L. Kronig.  In molecular gases, Kronig, Petersson, Hartree, and others correctly explained the phenomenon in terms of electron multiple scattering. In condensed matter, however, interpretation of the data was much less clear.  Various aspects of the phenomenon (e.g. accounting for thermal motion)  were included over approximately a 50 year period, but as late as the 1960's it was unclear whether or not long range order was essential to explain the phenomenon.
Around 1970 a collaboration between Edward A. Stern, Dale Sayers, and Farrel Lytle cracked the problem and demonstrated how EXAFS could be used as a quantitative tool for structure determination. 
The field has evolved significantly since 1970, and is advancing rapidly.

Schematic transmission experiment

(I/(Subscript[I, 0])) = (e^(-μ (E) x))μ (E) x=Ln(((Subscript[I, 0])/I))

Absorption edge energies are characteristic of the absorbing element. XAFS allows you to tune into different types of atoms by selecting the energy.

(Subscript[E, n]) ≃ ((-(Z^2) Rydberg)/(n^2))

Absorption coefficient

X-ray absorption probability can be calculated using standard quantum theory. As in optical spectroscopy, the absorption coefficient is proportional to the square of the transition matrix element, here shown in dipole approximation.

μ ∝  | < f| (Overscript[(ϵ ),  ⇀]) · 
          (Overscript[(r ),  ⇀]) |i > ((|)^2)

Δl = ± 1 (dipole approx)

K,L,M Absorption edges and selection rules

K:  1s -> p
(Subscript[L, 3]): 2(Subscript[p, 3/2]) -> s,d
(Subscript[L, 2]): 2(Subscript[p, 1/2]) -> s,d
(Subscript[L, 1]): 2s -> p

Useful Approximations

Between edges, absorption approximates a power law (straight lines on log-log plot).  Absorption coefficient decreases roughly as 1/(E^3).

K-edge energies for hydrogenic atoms go as (Z^2).  The K-edges of real atoms go as (Z^2.17)to a good approximation.

(Power law behavior of absorption edge energies vs Z)

\(K - \(edge : 
E\)\  = \ c*Z^p

      c	0.0059906

      p	2.172

      R	0.99997
\)

\(L1\  - \ \(edge : 
E\)\  = \ c*Z^p\)\(c	0.00020012\)\(p	2.559\)\(R	0.99998\)

\(L2\  - \ \(edge : 
E\)\  = \ c*Z^p\)\(c	0.00012731\)\(p	2.6549\)\(R	0.99993\)

\(L3\  - \ \(edge : 
E\)\  = \ c*Z^p\)\(c	0.00018468\)\(p	2.5434\)\(R	0.99965\)

The XAFS phenomenon

Here are some atoms in an octahedral configuration

outgoing s-wave (animation)

This animation is supposed to represent an outgoing s-symmetry electron wave.  There is no scatterer, just an outgoing wave, and no interference effect.

outgoing s-wave with scatterer (animation)

spherical p-wave (animation)

This animation is supposed to represent the outgoing p-symmetry electron wave.  The wave amplitude is zero in the direction perpendicular to the x-ray electric polarization vector, which gives a Cosine squared angle dependence for oriented samples.

Explanation of standard EXAFS equation

Conservation of energy relates electron wave number to x-ray photon energy by:

E - (Subscript[E, 0]) =  (ℏ^2) (k^2)/2 m

Several simplifying assumptions allowed Stern, Sayers, and Lytle to derive the following expression, the "standard EXAFS equation".  The damped sine wave form makes fourier transformation useful. It exhibits the essential features that permits a quantitative analysis of EXAFS.

Information available from XAFS

Average distance and mean square variation in distance
Coordination numbers, atomic species

EXAFS data reduction and data analysis

Steps in traditional EXAFS data analysis:
1) correction for instrumental effects
2) normalization to unit edge step
3) interpolation to k-space
4) background subtraction
5) weighting, k-windowing, and fourier transform
6) r-windowing and inverse transform to isolate single shell data
7) analysis of single shell amplitude and phase

Types of systems for which XAFS is applicable

XAFS is applicable to condensed matter (both crystalline and amorphous), and molecular gases.  Special conditions are not required - it is very suitable for in situ studies under real-life conditions.  If a sample is oriented on a molecular or crystalline level, polarized EXAFS can provide information on 3D structure.
XANES can provide information about site symmetry, bond lengths, and orbital occupancy.

Limitations of and extensions to the EXAFS equation

Spherical waves

large disorder

multiple scattering

multielectron excitations

Elementary Interpretation of XANES

2: Experimental design and methods I

Source requirements for XAFS

tunability  -  energy scans ~ 1 KeVbandwidth - ((~ 10)^(-4)) (depends on core hole lifetime)flux  -   >  (10^6)/secspectral purity  -  harmonics  <  .1  %spot size  -  depends on sample

Source Characteristics

{see notebook - synchrotron G function}

bend magnets
wigglers
undulators

Optics

monochromator types
	double crystal monochromator - 
		lambda = 2 d Sin(theta); E = hc/2d Sin(theta)
	variable exit - scan table to track beam motion
	fixed exit - translate crystals internally to monochromator
	sagittal focussing - bend second crystal to horizontally focus beam

harmonic rejection
	crystal cuts (Si(111),Si(220),etc.)
{see notebook - Diamond Structure Factor}
	detuning crystals
	mirrors

Cross sections

photons absorbed/time = cross section * (intensity)

mu = Sum(rho_i*sigma_i)

Experimental modalities

transmission (scanning and dispersive)
	geometry
	detectors
fluorescence
	geometry
	detectors
	filters
	other devices
total external reflection fluorescence
electron yield
Polarized XAFS

How to minimize common experimental problems (HALO)

Harmonics
Alignment
Linearity
Offsets

Thickness/particle size effects

Sample requirements, preparation, sample cells

{see notebook - thickness effects}

choice of fill gases

3: Experimental design and methods II

Fluorescence mode
Amplitude corrections (fill gas etc)
Scattered Background
effective count rate
Experimental setup
Choosing a detector
Integrating detectors
Stern/Heald detector, PIN diodes, PMTs in current integration mode
Pulse counting detectors
Ge and Si detectors, APDs, PMTs
dead time corrections
using filters with Ge detector
Optimizing filters and slits
Q of filter
eta of slits
finding optimum
improved potential at the APS
Novel analyzers being developed
	multilayer analyzer
	Laue analyzer
Sample requirements, preparation, sample cells
Thick dilute limit
Thin concentrated limit
Problem with thick concentrated samples
use normal incidence geometry to improve
Total external reflection fluorescence
Survey of other methods: electron yield detection, reflectivity,

4: XAFS Theory I

Derivation of EXAFS equation
Dependence of amplitude and phase on Z
Importance of mean free path
Importance of single scattering plus focussing MS
Thermal disorder, Einstein and Debye Models
Disorder regimes

5: EXAFS Data Analysis I  - EXAFS and disordered systems

Data reduction
Fourier filtering do's and don'ts
Cancellation of window effects
Ratio method
error estimates
Nonlinear least squares fitting
Pros and cons of R-space vs K-space fitting
Small disorder
moderate disorder
large disorder
Other methods (parametric models, splice, regularization)

6: XAFS Theory II

7: Data Analysis II:  MS - XAFS and XANES

8: Recent developments, exotic methods,  and future trends

DAFS
Magnetic XAFS, X-ray MCD
Photoconductivity
X-ray Raman
Optical detection
Acoustic detection