Scattering and diffraction. Kinematic
model
Kinematic
model for
X-ray diffraction
A material
sample is primarily regarded as a distribution of electrons
which
is peculiar to each case. The interaction of electromagnetic radiations
with matter is modeled considering radiation as particles (photons),
or as waves. We are dealing here with a typical ondulatory
phenomenon. In fact, in addition to the well known experiment
of Max
von Laue, the original experiments (Friedrich, Knipping and
Laue, 1912, in CuSO_{4} crystals)
were the proof that X-rays are waves and that crystals are
structured in three-dimensional lattices with periodic distances in the
range of X-ray wavelengths.
Waves scatter whenever there is a
change
in the incident wave front due
to a discontinuity in the medium in which they propagate (an astronaut
does not see the "blue sky"). If the phase relationship between waves
scattered by the discontinuity remain constant, the waves combine in a
cooperative and coherent manner, producing interferences, known as
diffraction. However, in the scattering process the phase relations
occur randomly and are not maintained over time.
The
theoretical model of matter-wave interaction is provided by the four
Maxwell equations, the equation of charge continuity and the two
equations that characterize the materials (the electric and magnetic
polarization and the dielectric and permeability constants).
Although
this model describes the phenomenon macroscopically, it also applies to
the atomic scale for electronic distributions weakly bound in the
atom and which move at speeds lower than light. In addition,
the nuclei are regarded as massive, and therefore fixed (Born
approximation). Furthermore, it is supposed that the material sample
extends indefinitely, that its dielectric constant is independent of
time, that the permeability of the material is close to the unit, and
that both constants are homogeneous and isotropic.
Basically,
the kinematic conditions are summarized considering that the incident
wave is hardly modified by the material as it passes through.
Therefore, the scattered wave is seen as a small perturbation due to an
interaction of a very weak magnitude. More specifically:
- The incident beam is considered homogeneous,
covering the sample completely and crossing it straight and
unchanged, so that its incident intensity is constant. This incident
intensity only decreases slowly, and isotropically, by a
regular
absorption mechanism when crossing the sample. This condition is
violated if the crystal sample is perfect, because there would be
interactions between the diffracted and incident beams, leading to extinction,
multiple
diffraction, anomalous
absorption ...
- The scattering occurs at discrete points of the sample,
producing a diffracted beam at an angle 2θ
with respect to the incident. The diffracted intensity is a small
proportion of the incident intensity (Kirchoff approximations for
visible light, and Born approximations for the phenomenon of
scattering). The small magnitude of the energy and momentum transfers
from the incident beam to to the diffracted ones implies that coherence
is maintained throughout the sample and that the thermal
diffuse scattering (TDS)
is not considered.
- The scattering is considered elastic (no change in
frequency) and coherent, and therefore the phase relationships due to
different crossed paths are maintained (diffraction), and the
whole sample diffracts in phase in the direction of the incident beam.
Therefore, Compton
scattering and the possible excitation of
electrons to other energy levels (which would lead to effects of fluorescence
and anomalous
scattering) are ignored. The interactions between X-rays
and nuclei are also ignored.
- Coherence means that the detected intensity is the result
of composing different diffracted beams, taking into account their
respective phases. In the kinematic model, with the detector far from
the sample (in relation to its size: Fraunhoffer diffraction model),
the diffracted spherical wave can be seen as a plane wave at the
detector, with all rays being parallel.
- All these features have to be met and implemented into the
experimental devices. If these conditions are not met, we have to
neglect the dynamic effects, or we have to apply the
corresponding corrections. A critical feature in the fulfillment of the
kinematic conditions is the mosaicity,
which takes into account that the sample deviates
from a
perfect single crystal, and that it is formed by single
crystal
micro-blocks distributed randomly and not being highly
distorted.
Distribution
of single crystal micro-blocks in a crystal sample suitable
for
implementing the kinematic model. As in many other aspects of
Crystallography, we must reach a compromise... The orientations must
not be totally random, but without reaching the
perfect
alignment of the micro-blocks. This is why we say that the
sample must be "perfectly
imperfect" .
Kinematic
model for a
single crystal
Continuing with the above, for the
structural analysis of samples using kinematic X-ray
diffraction, we consider that:
- An atom consists of a "point" nucleus, inert to X-rays. As
the duration time of the experiment and of the interaction is very
large, as compared with the frequency of the incident X-rays, it is
considered that the distribution of the Z electrons around the
nucleus is continuous. And this is why we can talk
about distribution
of electron density per unit of volume, ρ.
- This electron distribution can be
considered spherical (where the atomic scattering
factors do
not depend on the orientation of the diffracted beam, but only on the
angle θ),
or non-spherical (where the atomic scattering factors do depend on the
orientation of the diffracted beam)
when we take into account the distortions which occur in the electron
density of the valence electrons due to the interatomic bonds, the
free-electron pairs, non-spherical orbitals, intermolecular
interactions... In fact, X-ray diffraction experiments (sometimes
together with the neutron diffraction) can detect these density
deformations due to the electronic interactions.
- The electrons distributed around the nucleus are
considered weakly linked to the atom, so their vibrational
frequencies are very different from the incident radiation. If this is
not assumed, several corrections must be applied to the atomic
scattering factors to take into account the intrinsic phase
changes produced by the interaction with the linked electrons.
- In a single crystal, the distribution of the electron
density is periodic in the three dimensions of space. Therefore, one
only needs to know the electron density located inside the unit cell.
This model of periodic structure explains the correlations that define
the crystalline order.
- The unit cell (a simple mathematical representation of the
independent repeating blocks of the structure) contains the electron
density with concentrated peaks in a finite set of positions where it
is assumed that the atoms are located. Subsidiary maxima occur due to
the density distortions mentioned above. This electron density
distribution can also show different types of symmetry: exact
(crystallographic) or approximate (non-crystallographic). The
crystallographic symmetry, together with the translations that define
the lattice, define the so-called space group of the crystal
structure. All this means that the entire crystal sample can
be
described by repeating the independent part using the lattice
and
the symmetry of the space group. The independent part is known as the asymmetric
unit of the structure, and contains the motif that is
being repeated within the crystal.
- The experimental diffraction pattern obtained, and
therefore the structure that can be derived, represent an
average over the duration time of the experiment and over all
the
unit cells in the sample. That is:
<structure>
(over time and
unit-cells) =
<lattice
& symmetry> (according
to distortions, vibrations ...) +
<motif> (according
to vibrations, orientations, disorder ...)
(symbols
< > mean "average")
- All the components producing the averages arise from
the
fact that crystal samples are not perfectly ordered; they contain
a certain degree of disorder. The disorder can be static or dynamic.
Static disorder is due to the fact that repetitions are not always
exact; there is some mosaicity as well as modulations and some
inconmensurate crystal structures may occur as a consequence of small
shifts in the atomic positions. Dynamic disorder comes from the
different modes of thermal vibration of atoms around their equilibrium
positions; lattice phonons and harmonic or
anharmonic vibrations of the atoms or of the
molecule as a
whole.
- Finally, it must be remembered that the sample has a finite
size and a certain shape. Both characteristics have to be
taken
into account for some corrections not considered in the kinematic model.
But let's go
back...