Explain the statement “crystal act as three dimensional grating with X-rays”. Monochromatic X-ray beam of wavelength $λ=5.8189 A^0$ is reflected strongly for a glancing angle of $Ѳ = 75.86^0$ in first order by certain planes of cubic of lattice constant $3A^0$. Determine Miller indices of the possible reflecting planes.

Mumbai University > First Year Engineering > Sem 1 > Applied Physics 1

Marks: 7M

Year: May 2016

Crystal act as three dimensional grating with X-rays

1. Since the wavelength of X-rays is in the order of 1 A^0 or 〖10〗^(-8) cm, ordinary grating which has 6000 lines per cm cannot produce an appreciable diffrac-tion pattern of X-rays.
2. Therefore, in the case of X-rays, instead of ordinary grating crystals gratings are used. In crystal grating atoms are arranged at lattice points in a regular fashion.
3. These arranged atoms correspond to grating lines and the distance between two atoms is the grating element, in the order of le cm.
4. The crystal grating differs from optical grating in such a way that in crystal grating, the atomic centers are not in one plane but are distributed in 3-dimensional space. But in optical grating, they are limited to one plane.
5. Hence, crystal act as three dimensional grating with X-rays

Problem

Given:

$λ=5.8189 A^0$

$Ѳ = 75.86^0$

Lattice constant = $a = 3A^0$

Find:

Miller indices

Solution:

According to Bragg’s law,

$λ= 2d sin Ѳ$

$5.8189= 2d sin 75.86$

$d=3.0003 A^0$

Now, for a cubic crystal the interplanar spacing d is related to the lattice constant a by the equation,

$d = \frac{a}{\sqrt{(h^2+k^2+l^2)}}$

$\sqrt{h^2+k^2+l^2} = \frac{a}{d} = \frac{3}{3.0003} = 0.99 ≈ 1$

This is possible only when one of h, k and l are equal to 1 and other two equal to zero.

Thus, the Miller indices for the reflecting planes are (1 0 0), (0 1 0), (0 0 1)