# In X - Ray diffraction analysis the angle of the reflected beam for 1st order diffraction is sf(21.9^@). If the incident X - ray has a wavelength of 1.57color(white)(x)stackrel(@)"A"what is the separation of the crystal planes ?

Dec 12, 2015

$2.10 \textcolor{w h i t e}{x} \stackrel{\circ}{\text{A}}$

or $0.210 \textcolor{w h i t e}{x} \text{nm}$

#### Explanation:

This question is based on X - ray diffraction.

The incident X - ray beams are reflected off the crystal layers as shown in the graphic.

After reflection the beams interfere with each other to produce a pattern which can be detected with a photographic plate.

If the X - rays are in phase constructive interference occurs:

This produces a pattern on the plate enabling $\theta$ to be measured.

The condition for constructive interference to occur is that the path difference of the two beams must be a whole number of wavelengths.

From the graphic you can see that this path difference is $\mathrm{ds} \int h \eta$ going in + $\mathrm{ds} \int h \eta$ coming out.

Since this must be equal to a whole number of wavelengths $\lambda$ we can write:

$n \lambda = 2 \mathrm{ds} \int h \eta$

Where $n$ is an integer.

In this case $n = 1$ so we can write:

$d = \frac{\lambda}{2 \sin \theta}$

$\therefore d = \frac{1.57}{2 \times 0.373} = 2.10 \textcolor{w h i t e}{x} \stackrel{\circ}{\text{A}}$

The stackrel(@)("A")"ngstrom" unit is not strictly SI. This is equal to $0.210 \textcolor{w h i t e}{x} \text{nm}$