Klein 314-317
Nesse 168
Crystal structure may be defined as the orderly arrangement of atoms in
three dimensional space. Included then, in the definition is information on
How does one go about determining this information?
One cannot rely on the concept of inverse theory. This is a branch of
science that works on the premise that you know exactly how to mathematically
model physical phenomena. The observed data are then used to solve for the
physical parameters that describe the system. This approach unfortunately
does not work when using XRD data and the solving for the locations atoms
within a unit cell. We must therefore resort to a trial-and-error method
or forward modeling. When forward modelling, one assumes a certain model
solution is correct, and then tests to see how well the model result matches
the observations. Because there are many potential solutions to a problem,
optimization theory* is often employed to best-fit the model to the observed.
If the model result fits reasonably well, then the solution is assumed to
be correct. The "goodness of fit" between observe data and model solutions
is used as a criteria to determine if a solution is acceptable.
Diffraction patterns
Can we calculate the theoretical x-ray diffraction pattern of a crystal structure?
What determines the possible directions (i.e., possible angles)
in which a crystal can diffract a beam of monochromatic radiation?
Recall that diffraction can come from any number of (hkl) planes. Therefore, one needs an expression that will predict the diffraction angle for any set of (hkl) planes.
Starting with Bragg's Law:
There also exists geometric equations for each crystal system that relate the d-spacing of any given (hkl) plane and the lattice parameters.
For the Isometric system:
Let's work an example for halite.
a = 5.639Å. If (hkl) = (111), then a2 = 31.8 and (h 2+k 2+l 2) = 3.
1 / d2 = 0.0943. or d111 = 3.26 Å
Bragg's law and the equation above can be combined to give:
For a particular wavelength of monochromatic radiation (e.g., CuKa = 1.54059Å) and a particular crystal in the isometric system with a unit cell of length a (e.g. 5Å), then all possible Bragg angles of diffraction can be determined for every possible (hkl) plane.
Note that l 2 / 4 a 2 (e.g., 0.0237 using the above values) is always constant for any one diffraction pattern.
Also note that (h 2 + k 2 + l 2) will always be an integral value and certain combinations are not possible (i.e., 7, 15, 23, 28, ...).
For all possible families of planes hkl in a unit cell, it is possible to calculate the angle at which reflection would occur.
Example:
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In short, this tells us that diffraction directions are determined
solely by the shape and size of the unit cell.
Atomic Locations - Space Group effects
There are particular atom arrangements that reduce the intensity to zero.
The presence or absence of certain index lines is therefore related to the
Bravais lattice type.
The directions of diffracted beams (i.e., the location of (hkl)
reflections) can only tell use about the size and shape of the unit cell.
We need intensity information to tell us about the location of atoms within
the unit cell.
Atom Types - Atomic Scattering Factors
The intensity of scattered X-rays is related to the number of electrons
in the atom. Scattering efficiency is directly proportional the number of
electrons if the scattering is parallel to incident beams (2Q = 0°). This relationship gets more
complicated as angles of 2Q become higher.
As a general rule, Fe2+ will scatter more than Si4+,
which scatters more than O2-.
There are other scattering factors, however the important relationship
to remember is
Crystal Structure Diffraction Pattern
Three steps to determination of an unknown structure ---> the lattice
parameters and atom positions.