X-Rays
- Page ID
- 35135
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- Interpret the symbols used in the Bragg equation.
Like light, X-rays are electromagnetic radiation with very short wavelengths. Thus, X-ray photons have high energy, and they penetrate opaque material, but are absorbed by materials containing heavy elements.
X-ray Diffraction
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 interatomic 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 established technique for material analysis. This link is the home page of Lambda Research, which provide various services using X-ray diffractions. For example:
- Residual Stress Measurement
- Qualitative Phase Analysis
- Quantitative Phase Analysis
- Precise Lattice Parameter Determination
In 1913, the father and son team of W.H. Bragg and W.L. Bragg gave the equation for the interpretation of X-ray diffraction, and this is known as the Bragg equation.
\(2\, d\, \sin \theta = n\, \lambda\)
where d is the distance between crystallographic planes, \(\theta\) is half the angle of diffraction, n is an integer, and \(\lambda\) is the wavelength of the X-ray. A set of planes gives several diffraction beams; each is known as the nth order.
The X-ray wavelength from a copper X-ray is 154.2 pm. If the inter-planar distance from \(\ce{NaCl}\) is 286 pm, what is the angle \(\theta\)?
Solution
\(\begin{align*}
\sin \theta &= \dfrac{\lambda}{2 d}\\
&= \dfrac{154}{2\times282}\\
&= 0.273
\end{align*}\)
\(\theta = 15.8^\circ\)
An X-ray of unknown wavelength is used. If the inter-planar distance from \(\ce{NaCl}\) is 286 pm, and the angle \(\theta\) is found to be 7.23°, what is \(\lambda\)?
Solution
\(\begin{align*}
\lambda &= 2\, d\, \sin\theta\\
&= 2\times282\times\sin(7.23^\circ)\\
&= \mathrm{71\: pm}
\end{align*}\)
The X-ray of wavelength 71 pm is used. If the inter-planar distance from \(\ce{KI}\) is 353 pm, what is the angle \(\theta\) for the second order diffracted beam?
Solution
The calculation is shown below:
\(\begin{align*}
\sin \theta &= \dfrac{\lambda}{2 d}\\
&= \dfrac{71}{2\times353}\\
&= 0.100\\
\theta &= 5.8^\circ
\end{align*}\)
These examples illustrate some example of the applications of X-ray diffraction for the study of solids.
If the wavelength is 150 pm and the interplanar distance d is 300 pm, what is the angle \(\theta\) in the Bragg equation, for n = 2?
- Answer
-
30 degrees