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Relationship between crystalline structure and X-ray data: peak positions, intensities and widths

Peak positions

Using Bragg's Law, the peak positions can be theoretically calculated.

\[\theta = \arcsin \left( {\frac{\lambda }{{2d}}} \right)\]

For a cubic unit cell:

d = \(\frac{a}{{\sqrt N }}\), where N = h2 + k2 + l2 and a is the cell parameter.

(More complex relationships for less symmetrical cells are given in most standard text books)

So the measured value 2θ can be related to the cell parameters.

In the earlier video peak was observed at about 44.4. Knowing the wavelength, 1.54Å, and using Bragg's Law gives a d-spacing of ~2.04 Å. Some additional information is required to obtain lattice parameters from this d-spacing. Knowing that LiF has a cubic structure with a unit cell ~4.03, means that this reflection must be (002) (which is the same as (200) and (020)).

Peak intensities

The structure factor, Fhkl, of a reflection, hkl, is dependent on the type of atoms and their positions (x, y, z) in the unit cell.

\[{F_{hkl}} = {\sum\limits_i {{f_i}} _{}}\exp 2\pi i(h{x_i} + k{y_i} + l{z_i})\]

fi is the scattering factor for atom i and is related to its atomic number.

The intensity of a peak I hkl is given by:

\[{I_{hkl}} \propto {\left. {\left| {{F_{hkl}}} \right.} \right|^2}\]

The proportionality includes the multiplicity for that family of reflections and other geometrical factors.

Differences in intensity do relate to changes in chemistry (scattering factor). However, most commonly for multiphase samples, changes in intensities are related to the amount of each phase present in the sample. Suitable calibration factors are required to perform quantitative phase analysis.

Peak widths

The peak width β in radians (often measured as full width at half maximum, FWHM) is inversely proportional to the crystallite size Lhkl perpendicular to h k l plane.

\[{L_{hkl}} = \frac{\lambda }{{\beta \cos \theta }}\;\;\;\;\;\; \rm{Scherrer\;\; equation} \]

(Whilst small crystals are the most common cause of line broadening but other defects can also cause peak widths to increase.)

In the next section there is a simulation which shows how changes in the structure of a simple cubic material influences the diffraction pattern.

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