Exploration and Optimization of Tellurium‐Based Thermoelectrics

The closely related electron density (Equation 4.3 (b)) is associated with the structure factor via the Fourier
transform function, which allows for a convenient scientific relationship: If a crystal’s structure factors
are known, one can calculate the precise locations of electron density in the crystal and therefore, know
the precise location of the atoms and their sizes within. Diffraction is the affected X‐rays, in this case,
bending or modulating due to the presence of electrons. Electron density not only controls diffraction,
but provides information on the size of the atoms involved due to each element’s unique electron
density. Because the scattered X‐ray beam is reduced during diffraction as with the amplitude, a scaling
factor, , and two corrections – Lorentz, , (geometrical) and (polarization) – are added to Equation 4.4
displayed below, the experimental intensity is equivalent to the square of the wave’s amplitude or
structure factor .
∙ Equation 4.4 Diffracted intensity.
This provides a comparison between the observed reflections in reciprocal space and their
corresponding calculations determined from a structural model.
4.1. Powder X‐Ray Diffraction (p‐XRD)
Experimentally, a polycrystalline solid‐state sample is undoubtedly easier to obtain than a single
crystal of the same sample. That is, a sample that is comprised of small crystallites typically on the order
of 0.1 – 100 μm and randomly oriented, or stacked along specific orientations. When a monochromatic
beam of X‐rays hits a powdered sample, the diffraction no longer occurs in a set direction as discussed
above, but continues in all directions (that obey Bragg’s Law) as a Debye‐Sherrer diffraction cone (Figure
4.1) – named for the Dutch and Swiss physicists who, in 1915, first attempted the technique with a strip
of photographic film surrounding their sample as a collection method for the diffracted marks.
Figure 4.1 Debye‐Scherrer powder diffraction cone. [125]
41