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Powder Diffraction

A powder pattern is like a ‘spectrum’ of d-spacings in the crystal structure and is usually presented in the form of a line trace. A finely ground crystalline powder contains a very large number of small crystals known as crystallites, which are orientated randomly to one another. If such a sample is placed in the path of a monochromatic x-ray beam, diffraction will occur from planes in those crystallites, which happen to be orientated at the correct angle to fulfil the Bragg condition. The effect of this is that each lattice spacing in the crystal will give rise to a cone of diffraction. In fact, each cone consists of a set of closely spaced dots each one representing diffraction from a single crystallite within the powder sample. The diffracted beam makes an angle of 2θ with the incident beam.

$This is a schematic diagram showing how a powder diffractometer functions.$

Schematic diagram of a Powder Diffractometer.This equipment uses an x-ray detector (typically a Geiger-Muller tube or scintillation detector) to measure the positions of the diffracted beams.

In order to obtain powder x-ray diffraction data in a useful format necessary for analysis, the positions of the various diffraction cones need to be determined. This can be achieved by using photographic film or a detector sensitive to x-ray radiation. Both techniques allow us to determine the angle, 2θ, of the diffracted beam of the various diffraction cones.

Disadvantages of Powder Diffraction:

Indexing patterns is much more complicated in non-cubic systems.
Peak overlap may occur – solving crystal structure depends on being able to measure intensities for individual reflections, whereas this is not possible if there is overlap.
Overlap progressively gets worse for high angle reflections.

Uses of Powder Diffraction:

Finger printing and phase identification.
Measuring sample purity.
Confirming that the bulk sample is the same as the single crystal used for a structure determination.
Study of samples under extreme conditions and phase changes which occur due to temperature and pressure changes, (crystals under very high pressure may crumble to a powder).
Quantitative analysis.

Why crystallography?

Experimental Powder Diffraction Analysis

For a cubic lattice:

(1)

sin²θ =

4a²

(h²+k²+l²)

Hence:

(2)

a² =

λ²

4sin²θ

(h²+k²+l²)

Example (a):

2θ	sin²θ	(h²+k²+l²)	hkl	a
26.58	0.053	1	001	3.34
37.94	0.106	2	011	3.34
etc	-	-	-	-
etc	-	-	-	-

Important facts for powder diffraction analysis

(h²+k²+l²) must be an integer.
hkl: square each number to add up to the (h²+k²+l²) value.
Calculate a value of a for every row and then average to achieve final value.
Each value of (h²+k²+l²) is produced by taking ratios of the sin²θ figures.

eg: For (h²+k²+l²) = 2, the value 2 is produced as a ratio between 0.053 and 0.106.

NOTE: If (h²+k²+l²) is ever equal to 7, this is impossible!!!
Hence new ratios need to be calculated excluding this value.

Crystal structure determination from tabulated values:

Primitive = no condition.
I-centred = h+k+l = 2n only (an even total).
F-centred = h,k,l are all odd or all even.

Example.
(hkl): 111 [1+1+1 = 3 Odd, therefore not I-centred]
(hkl): 210 [2=Even, 1=Odd therefore not F-centred]

Hence for example (a) shown above, the crystal structure has no condition and is therefore PRIMITIVE CUBIC.

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