11 - Lecture notes for GEOL3010



Crystal Structure Determinations

Crystal structure may be defined as the orderly arrangement of atoms in three dimensional space. Included then, in the definition is information on

  1. Unit cell parameters (i.e., the size and shape of the unit cell).
  2. The location of atoms (implied then are set bond angles between atoms).
  3. Space group symmetry (i.e., the symmetrical relations of atoms in the unit cell).
  4. Chemical content of the units cell.

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.


*One common example of optimization theory is seen in the fitting of paired data arrays to an equation that describes a line of the form y = mx +b. If the correlation coefficient (r2) (an often cited parameter in statistics) is close to one, then the model line is said to fit the data. If r2 is near zero, then the model line has no bearing on the data trend . This is why we learn calculus (i.e., recall minimization occurs when dx/dy= 0).

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:

n λ = 2 d sin Θ

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 = 5.639.

If (hkl) = (111), then a2 = 31.8 2 and (h 2+k 2+l 2) = 3.

1 / d2111 = 0.0943    or     d111 = 3.26

 

Bragg's law and the equation above can be combined to give:

For a particular wavelength of monochromatic radiation (e.g., CuK α = 1.54059 or CoK α = 1.7890) and a particular crystal in the isometric system with a unit cell of length a (e.g. 5), then all possible Bragg angles of diffraction can be determined for every possible (hkl) plane.

Note that λ 2 / 4 a 2 (e.g., 0.0237 using the values CuKα = 1.54059 and a = 5) 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: Using CuKα = 1.54059

 Line

 sin2Θ

  (h 2 + k 2 + l 2)

λ 2 / 4 a 2

 a ()

 hkl

 1

 0.0712

 3

 0.0237

5.0

 111

 2

 0.0949

 4

  0.0237

 5.0

 200

 3

 0.1899

 8

  0.0237

 5.0

 220

 n

 ...

 ...

 ...

 ...

 ...

 

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 (2Θ = 0). This relationship gets more complicated as angles of 2Θ become higher. As a general rule, Fe2+ (24e-) will scatter more than Si4+ (10e-), which scatters similarly to O2- (10e-) at 0. At higher angles  Si4+ 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.

  1. The size and shape of the unit cell can be deduced from the positions of lines. Assume a crystal class and assign the (hkl) reflections. This is called indexing the pattern.
  2. Once the size and shape of the unit cell is known, then the number of atoms per unit cell can be calculated from chemical data and its measured density.
  3. Finally, the positions of the atoms within the unit cell are deduced from the intensity measurements. This is called a unit cell refinement.