There are a couple of other techniques which we should
mention. The first one is x-ray scattering from surface
.
This technique
is very similar to bulk x-ray scattering with the major advantage over
LEED that the theory is purely kinematic. The problem of the long mean
free path of the x-rays is overcome by using a grazing incidence
angle. In such a geometry total external reflection occurs
because the refractive index for x-rays is smaller than
one. It is, however, only very little smaller than one (in the order
of 
such that the incidence angle below which total
reflection occurs is very small, a few tenth of a degree. Therefore,
surface x-ray scattering requires a highly collimated x-ray beam
which can only be produced at a synchrotron radiation
source delivering
high-brilliance light.
It is interesting to compare the advantages and disadvantages of LEED and surface x-ray scattering. One is that the momentum transfer, and hence the sensitivity to structural parameters, is nearly perpendicular to the surface in case of LEED and nearly parallel to the surface in the case of x-ray scattering. The penetration depth is a little higher for x-ray scattering but it has the same order of magnitude as for LEED.
There are several interesting experiments involving the scattering of
atoms and ions but we are not going to discuss them here (apart from
He scattering in the section about vibrational properties, section
). If you are interested in these techniques,
consult the book by Woodruff and Delchar, section .
Two techniques based on electron scattering are discussed here in further detail: (S)EXAFS and photoelectron diffraction. Both do not rely on long-range order on the surface (like LEED). Instead, they can be used to determine the structure locally around an atom of interest. This is achieved by using specific atoms as ``electron sources'' instead of an external source like in LEED.
The techniques of EXAFS and SEXAFS have been made possible by the construction of synchrotron radiation (SR) sources. SR has a continuous energy spectrum. In combination with a monochromator it provides a tunable x-ray source. The (S)EXAFS measurements involve scanning the photon energy around the absorption edges of the atoms in a material or on a surface. The fine structure in the absorption cross section gives information about the neighbours of the emitting atoms. The main advantages of these x-ray absorption techniques are that they work for materials where long-range order is not present and that at least the nearest neighbour distances can be obtained with rather high precision.
Fig.
shows an example for an EXAFS spectrum.
Such a spectrum can be taken
by exposing a thin film of material to x-rays and simply measuring the
transmission through the film.
The x-ray
absorption of Cu is plotted vs the photon energy. As the energy
reaches the K-edge the absorption increases steeply because of the the
possibility to excite the K-electrons. Above the edge the absorption
shows a slow decrease due to the matrix element. On this slow
decrease rapid oscillations in the cross section are superimposed.
These are the EXAFS oscillations.
Figure: X-ray absorption of Cu in the vicinity of the K-edge.
. The
initial state is simply the localized core state. The final state is
the extended state of the outgoing electron wave, including all the
multiple scattering
processes. One can now think about the EXAFS
oscillations in the following way. At low kinetic energies, from zero
to a few hundred electron volts, the cross section for the
back-scattering of the electrons from the neighbour atoms is rather
high. These back-scattered waves have to be added coherently to the
outgoing wave and this directly influences the final state at the
emitter and thus the matrix element. The interference from outgoing
and back-scattered waves changes with a periodicity given by the
nearest neighbour distance. The effect is illustrated in Fig.
.
Figure: Schematic illustration of the interference leading to
the EXAFS oscillations
This simple picture works only for electron energies which are not too close to the edge. Below kinetic energies of 50 eV or so the oscillations contain resonant absorption from the valence states.
It is convenient to extract the EXAFS oscillations from the slowly
varying background. This is done by the definition of a ``fine
structure function'' 
where 
is the measured absorption and 
is the
absorption due to the free atom.
If we consider a simple singe-scattering picture the fine structure function is given by
where k is the electron wave number, 
is an amplitude function
defined below and 
is a phase shift. The sum runs over
different ``shells'' of neighbours, a shell being defined as a set of
neighbours having the same distance from the emitter, 
. In principle,
the desired values for can be extracted from
by a Fourier transformation. The phase shift would not be a problem in
such an analysis but its energy dependence is one. It leads
to wrong values for . Thus the phase shifts
have to be included in
the analysis. One can use calculated phase shifts or one can, in
contrast to LEED and Photoelectron Diffraction, use ``experimental''
phase shifts. The reason is that only the phase shift for
back scattering is of interest, not the phase shifts for
all the other scattering angles. The phase shift can be
obtained from a material which contains the scatterer of interest and
has a known structure, e.g. a single crystal.
Let's look again at the amplitude function for the different shells. It is given by
is the number of atoms in the shell. The 
factor
leads to an effective localization of EXAFS explaining the success of
a single-scattering treatment. It is caused by the fact that both
emitter and scatterer are point sources. The next factor is the
modulus of the scattering amplitude. W(T,K) is a Debye-Waller factor
which takes the thermal vibrations into account and the last factor
describes the inelastic scattering of the electrons in the solid. As
mentioned above, the single scattering approach works quite well, at
least in order to determine the distance of the nearest neighbour shell.
The surface version of EXAFS is called SEXAFS. One can for example use
the technique for determining the bond distances of adsorbate atoms to
the substrate. However, what one actually wants to know is the adsorption site
and this is difficult to get from just the bondlength. Help could come
from considering the absolute amplitude of the modulations because
this should give the number of nearest neighbours (equation ).
But due to experimental
difficulties (see below) it is very dangerous to use the absolute
modulation strength. One can also play the following trick. The synchrotron radiation
one uses for the experiment is polarized. This means that the
electrons from a core level will have a certain angular distribution,
depending on the direction of the polarization vector. One can take
SEXAFS data for different directions of the polarization vector such
that different possible neighbours would be ``hit'' by a high intensity
of photoelectrons. Comparing the different SEXAFS spectra one can
work out where the nearest neighbours are.
One has to add a few words of caution, though. SEXAFS is a very difficult experiment. The first problem is what to measure in order to get the absorption of a particular atomic species on the surface. One possibility is the intensity of the Auger signal which is emitted by the decay of the core electron one creates. Such Auger peaks are usually positioned on a high background of inelastically scattered electrons, leading to a bad signal-to-noise ration. To make matters worse, the SEXAFS modulations are only a tiny fraction of the absorption cross section (one percent or so) and there are only a few adsorbate atoms compared to a bulk EXAFS experiment.
We just give one example here which demonstrates how powerful SEXAFS
is as a technique for structural investigations. SEXAFS has be used to
determine the structure of the co-adsorption system SO 
+ O on
Cu(111) which is formed upon SO adsorption [36].
Fig. shows data taken at the oxygen K-edge
for two different polarizations of the incident photons. The spectra
are quite different, illustrating the the usefulness of the
``trick'' mentioned above. The Fourier transforms of the data are
shown together with the result of a simulation for a particular
geometry. The agreement for the closer distances is very good.
Figure: SEXAFS data from the oxygen K-edge of the co-adsorption
system SO + O on Cu(111). After ref. [36].
Fig.
shows the final result of this structural
determination showing the position of the adsorbates and
a very complex surface reconstruction.
Figure: Structure SO + O on Cu(111) as determined by SEXAFS.
After ref. [36].
