DIFFRACTION FROM CRYSTAL PLANES
The only requirement for diffraction is that
constructive interference occurs i.e. that the path
length difference is a whole number of
wavelengths.
This is usually illustrated so that:angle of incidence = angle of reflection (Braggs
Law)
This is because this gives the highest signal.
Thus, in reality the incidence angle and reflected
angle are scanned together maintaining this
equality of angles. At various specific angles the
Bragg law will be satisfied for a particular crystal
plane.
Two reflections from different planes
Lets imagine one piece of solid. That satisfies
Bragg reflection as shown.
We will observe strong diffraction
But if the planes are misaligned
We will observe no diffraction
In any real solid we have a chance orientation. It
would be almost impossible to study diffraction.
However, 99% of all materials are polycrystalline
or can be prepared (by grinding) so as to present
many grains of material. In these some will always
be at the correct alignment.
Those planes at the right orientation will give
strong diffraction
Basis of POWDER X-RAY DIFFRACTION
EXPERIMENTAL SET-UP
Modern machines rotate the x-ray source and the
detector.
detector
source
sample
X-ray sources are cumbersone. To generate
sufficient signal need to generate lots of power i.e
x-rays. Frequently run at 50 kV and 50 mA.
anode
e’s
5V
-ve
50 kV +ve
filament
Moving these are difficult because of the electric
supplies and water used to cool anode to prevent
melting. Older machines moved detector through
2Θ and sample through Θ. This maintains angle
incidence = angle of reflection.
For historical reason we plot intensity versus angle
2Θ
X-RAY LINE BROADENING
When x-rays enter a solid they undergo refraction.
For x-rays this is very small.
But the refraction angle differs from the incidence
angle by only parts per thousand. But because of
this the path length difference slowly varies from
planes deeper into the material. The constructive
interference slowly becomes destructive.
Provided the sample is thick enough (if a sample is
10 μm there are 1 x 10-5/10-10 atom planes = 105)
then all these slightly out of phase reflections will
cancel. Leave just the perfect constructive
interference feature. Shown by rocking curves.
Scan detector across the diffracted beam.
2500
intensity
2000
1500
1000
500
0
44.8
45
45.2
45.4 45.6
45.8
46
46.2
angle 2-theta
Called rocking curve – it is a measure of how
crystalline a material is. A Si wafer rocking curve
could have a rocking curve of 0.1 mrad.
If the samples are thin incomplete cancelling is
observed. The diffracted peaks are not narrow they
become broad. The broadness can be used to
estimate the sample thickness.
Formulism was drawn up by Scherrer:Sample thickness, t
= 0.9 λ/(B cos θ)
B = Bactual = √(B2obs – B2o)
where Bobs = FWHM of reflection
Bo = instrumental FWHM minimum