Skills to develop
- Explain X-rays
- Interpret the symbols used in Bragg equation
Like light, X-rays are electromagnetic radiation with very short wavelengths.
Thus, X-rays photons have high energy, and they penetrate opaque material,
but absorbed by materials containing heavy elements.
When light passes through a series of equal-spaced pinholes, it gives rise
to a pattern due to wave interference, and such a phenomenon is known as
diffraction. X-rays have wavelengths comparable to the inter
atomic distances of crystals, and the interference patterns are developed
by crystals when a beam of X-rays passes a crystal or a sample of
crystal powder. The phenomena are known as diffraction of X-rays by
crystals. More theory is given in
Introduction to X-ray Diffraction.
X-ray diffraction, discovered by von Laue in 1912,
is a well establised technique for material analysis. This link is the home
page of Lambda Research, which provide various services using X-ray
diffractions. For example:
In 1913, the father and son team of W.H. Bragg and W.L. Bragg gave the equation
for the interpretation of of X-ray diffraction, and this is known as
the Bragg equation.
2 d sin q = n l
where d is the distance between crystallographic planes, and
q is half the angle of diffraction,
n is an integer, and l is the wavelength of
the X-ray. A set of planes gives several diffraction beams, each is known
as the nth order.
- Residual Stress Measurement
- Qualitative Phase Analysis
- Quantitative Phase Analysis
- Precise Lattice Parameter Determination
An animated illustration of Bragg equation shows the graphical
relationship of the variables in the equation. The units used for wavelength
and distances are &176; and 1 &176; = 100 pm.
The X-ray wavelength from a copper X-ray is 154.2 pm. If the inter-planar
distance from NaCl is 286 pm, what is the angle q?
sin q = l / (2 d)
= 154 / 2*282
q = 15.8°
The X-ray of unknown wavelength is used. If the inter-planar distance
from NaCl is 286 pm, and the angle q is found to
l = 2 d sinl
= 71 pm
The X-ray of wavelength 71 pm is used. If the inter-planar distance
from KI is 353 pm, what is the angle q for the
second order diffracted beam?
sin q = l / (2 d)
The calculation is shown below:
= 71 / 2*353
q = 5.8°
These examples illustrate some example of the applications of X-rays
diffraction for the study of solids.
Confidence Building Questions