RMC
|
EXAFS:
Extended X-ray Absorption Fine Structure |
|
|
Introduction
XAS
(X-ray absorption
spectroscopy) is an element specific method to investigate the bond angles, bond
lengths and coordination numbers. During the experiment the material under
investigation is targeted with monochromatic X-ray beam (produced by
synchrotron radiation). Some of the X-ray photons are absorbed by the material,
and the rate of the absorption is measured versus the X-ray photon energy.
The
X-ray absorption coefficient of a homogenous material can be given by
|
|
where E is the photon energy and d
is the thickness of the sample. Generally, the absorption of X-ray is decreasing with the increasing
energy of the X-ray photons, but distinct spikes corresponding to a drastic
increase of the absorption can be detected at some energy. These are the absorption edges, and they correspond to the binding
energies of the inner-shell electrons (K, L, M..). As each chemical element has
specific, well-defined binding energies, it is possible to select an energy
range for the X-ray beam sweeping specifically only an absorption edge (and the
following) region of a selected element. This way information of the
neighbourhood of the atoms of this chosen chemical element can be obtained.
XAS spectrum can be divided into different parts based
on the energy range of the X-ray beam compared to the absorption edge, but
there is no consensus in the literature neither in the number nor in the
limiting energy values of the ranges. A
possible division is given here.
1.)
Directly before the absorption edge can be found the pre-edge region, where no ionisation occurs, only
transition to higher, non-completely filled or empty orbits.
2.)
In the edge region, where photon energy, E £ E0+10 eV (E0
is the ionisation energy) the XANES (X-ray Near Edge
Structure) is observable.
3.)
NEXAFS (Near-Edge X-ray Absorption Fine Structure) region is
between E0+10<E £ E0+50 eV.
4.) EXAFS
region is where E >E0+50
eV.
It has to
be noted, that sometimes there is no division between the XANES and NEXAFS
region, and the two acronyms are used as synonyms, although NEXAFS is usually
used in connection with organic molecules and surfaces. The absorption edge
itself sometimes is not considered to be in the XANES region, and the beginning
of this region is set at E > E0+5 eV. Sometimes the
division between the XANES-NEXAFS and EXAFS region is set around 150 eV. From
now on the acronym XANES will be used for the description of the XANES-NEXAFS
range. A typical XAS spectrum can be seen in Figure
1.
Figure 1: XAS spectrum to show the
different methods depending on the energy range of the X-ray photon. m(E) is the energy dependent absorption given by E 1, the energy is given in eV.
The structure of the XAS
spectrum
When the
X-ray photon energy reaches the ionisation energy of an inner shell atom, the
absorption rapidly increases creating the ascending part of the absorption edge
peak. After the maximum the decrease of the absorption is not smooth, but
larger and smaller wave crests and troughs are super-positioned on the
exponentially decreasing signal. This is the so-called XAFS (X-ray
Absorption Fine Structure), which can give statistical information about the local structure around
the absorber atoms. The reason for the XAFS is the following. XAFS appear only
if there are atoms near the absorbing atom, so in solid and liquid state and in
molecular gasses. In case of free atoms as for example noble gasses no XAFS can
be observed.
When the
X-ray photon with E>E0 is absorbed, its whole energy is
transferred to an inner-shell electron, which will jump to unoccupied continuum
level with symmetry properties defined by the dipole selection rules. If the
configuration interaction is negligible, than the selection rule is Dl=±1. The absorption edges are denoted according to the shell of the
ejected core electron (K, LI,
LII, LIII). The ejected photoelectron will
have kinetic energy E-E0. The outgoing photoelectron wave
can be scattered by the neighbouring atoms, and the XAFS is caused by the
interference of the outgoing and backscattered electron wave at the position of
the absorber atom.
Initial state |
Final
state |
|
|
K |
2S1/2 |
p |
|
LI |
2S1/2 |
p |
|
LII |
2P1/2 |
d, (s) |
|
LIII |
2P3/2 |
d, (s) |
There is
however a difference in the physical phenomenon taking place at low or higher
kinetic energy, In the XANES region, where the kinetic energy of the photoelectron
is low, multiple scattering of the photoelectron
by the neighbouring atoms occurs. This is due to the fact, that the
corresponding wave vector of the photoelectron is large enough to be comparable
to the inter-atomic distances between the absorber and its neighbours, and the
scattering amplitude is large. Due to the multiple-scattering process, XANES
probes the 3 and more-body correlations and can give information not just about
bond lengths (or inter-atomic distances), but about the bond angles, too.
In the
EXAFS region the kinetic energy of the photoelectron is higher and consequently
the wave length and the scattering amplitude is smaller, so mainly single
scattering of the photoelectron by the neighbouring atoms takes place.
The scattering amplitude and phase shift caused by the
backscatterer depends on of the neighbour’s type, and the phase and the
amplitude of the backscattered wave depends on the inter-atomic distance
between the absorber and the backscatterer as well.
After
the ejection of the photoelectron the absorber atom will be in an exited state
due to the core hole. Relaxation can occur when an electron occupying a higher
energy level jump down into the core hole. The energy difference between the
levels is either emitted as a fluorescence photon (only rarely occur), or
absorbed by an other higher-level electron, which is emitted from the atom
(Auger-electron).
The
wave vector of the photoelectron (k) can be calculated from the X-ray
photon energy, E and the ionization energy, E0 by:
The
oscillating absorption coefficient is normalized by the smooth atomic
absorption background, m0 defining the EXAFS signal, c(k):
RMC specific remarks
It has
to be noted, that in RMC c2 is applied for denoting the
difference between the experimental and calculated data, so it should not be
confused with c(k). For fitting EXAFS data in RMC,
the values of k and c(k) has to be given in the EXAFS
data file as the input experimental EXAFS data.
The
EXAFS experimental data used in the fitting is denoted by E(k) and it is
calculated by 
after reading the k and c(k)
data from the experimental data file. The weighting factor n has to be
given in the *.dat file, and it is a small integer usually between 1 and
3. The reason for using this weighting is to compensate for the amplitude
decay, fitting with n=1, 2, 3 results in a physically more realistic
configuration than in case of n=0, kmax is the largest
k value of the data set.
The
coefficients, c(r,k) for the Fourier-transformation of the r-space
information to k-space are r,k dependent, and has to be given in
the coefficient file. The coefficients can be calculated for example by program FEFF, see some information about it here. Care has
to be taken, that the same r-points have to be used during the
coefficient calculation as used in the RMC simulation. Keep in mind, that in
RMC always the middle of the bin is used for the given bin! It is possible to
use only a range of the r-dependent coefficients given in the coefficient file,
this can be specified in the *.dat file. Different ranges can be
specified for the different partials (all the possible atom type pairs of
the edge particle type) for an absorption edge. The minimum and maximum indices of the r points (columns) for the partials are given in the *.dat file, in the
order rmin1, rmax1, rmin2 rmax2,
->rminN, rmaxN, where the indexing refers to the
partials contributing to the given edge. For example for a 3-component system
of As, Se, I for the As edge there are three partials (As-As, As-Se, As-I), for
which the r index data has to follow in RMC order for the partials.
The
calculation of EC(k) data is performed according to the
following formula:
where H(ip,ir) is the value of the ip-th
partial histogram ir-th histogram bin, Ne is the number
of the particles with the absorption edge. The first sum is going through all
the partials containing the particle type producing the absorption edge.
The squared difference, 
is calculated as
only using constant and
multiplication factor for renormalizations.
Links
There
are plenty of more detailed descriptions of EXAFS in the Internet, for example:
http://en.wikipedia.org/wiki/EXAFS
http://www.ieap.uni-kiel.de/surface/ag-skibowski/xanes/xanes.htm
http://www.p-ng.si/~arcon/xas/exafs/exafs.htm
http://gbxafs.iit.edu/training/XAFS%20intro.pdf
Last modified on 19.09.2012 by Orsolya
Gereben