Question

Solved Problem. The first order reflections from the 100,
110 and 111 planes of a given cubic crystal (NaCl Crystal) were
found to occur at angles 5.9°, 8.4° and 5.2° respectively.
Determine the type of cubic lattice to which the crystal belongs.​

Answer 1

Answer:

The crystal is face centered cubic centered.

Answer 2

Answer:

The type of cubic lattice is a face-centred cubic system

Explanation:

As per the question

Reflection from NaCl planes 100, 110, and 111 at an angle of 5.9°, 8.4° and 5.2° is of the first order.

According to Bragg's law

$\[n\lambda = 2d\sin \theta \]$

For all the planes wavelength of light is the same therefore $\lambda$ is constant

$\[\frac{1}{d} \propto 2\sin \theta \]$

For plane 100

$\[\begin{gathered} \frac{1}{d} \propto 2\sin \theta \hfill \\ \frac{1}{{{d_{100}}}} \propto 2\sin 5.9^\circ \hfill \\ \frac{1}{{{d_{100}}}} \propto 0.205585 \hfill \\ \end{gathered} \]$

For plane 110

$\[\begin{gathered} \frac{1}{d} \propto 2\sin \theta \hfill \\ \frac{1}{{{d_{110}}}} \propto 2\sin 8.4^\circ \hfill \\ \frac{1}{{{d_{110}}}} \propto 0.292166 \hfill \\ \end{gathered} \]$

For plane 111

$\[\begin{gathered} \frac{1}{d} \propto 2\sin \theta \hfill \\ \frac{1}{{{d_{111}}}} \propto 2\sin 5.2^\circ \hfill \\ \frac{1}{{{d_{111}}}} \propto 0.181265 \hfill \\ \end{gathered} \]$

Now, calculating the ratio of $\[{d_{100}}{\text{, }}{d_{110}}{\text{, and }}{d_{111}}\]$

$\[\begin{gathered} {d_{100}}:{d_{110}}{\text{:}}{d_{111}} = \frac{1}{{2\sin 5.9^\circ }}:\frac{1}{{2\sin 8.4^\circ }}:\frac{1}{{2\sin 5.2^\circ }} \hfill \\ {d_{100}}:{d_{110}}{\text{:}}{d_{111}} = 1:0.704:1.155 \hfill \\ \end{gathered} \]$

This calculated ratio is almost equal to the experimental ratio of the face-centred cubic system.

Therefore, the type of cubic lattice is a face-centred cubic system.