Question

Question: Analyzing crystal diffraction is intimately tied to the various different geometries in which the atoms can be arranged in three dimensions and upon their differing effectiveness in reflecting waves. To grasp some of the considerations without too much trouble, consider the simple square arrangement of identical atoms shown in the figure. In diagram (a), waves are incident at angle with the crystal face and are detected at the same angle with the atomic plane. In diagram (b), the crystal has been rotated 45⁰ counterclockwise, and waves are now incident upon planes comprising different sets of atoms. If in the orientation of diagram (b), constructive interference is noted only at an angle, $\theta =40°$at what angle(s) will constructive interference be found in the orientation of diagram (a)? (Note: The spacing between atoms is the same in each diagram.)

Short answer

Answer:

The constructive interference will be found in the orientation of diagram (a) at $\theta =27.0°and\theta =65.4°$.

Step-by-step solution

Bragg’s Law

Bragg’s Law states that $\mathbf{2}\mathit{d}\mathit{s}\mathit{i}\mathit{n}\mathit{\theta }\mathbf{=}\mathit{m}\mathit{\lambda }$, where d is the spacing between atomic layers.

Obtain angles

Here, we have two distinct setups for the same crystal. So, we have to use Bragg’s Law two times here. To connect both experiments, we should connect the distance between the atoms, which is equal to and also a constant, to the distance between the atomic planes which is equal to .

For setup (b), we have , here$\varphi =45°$ is the angle of planes. So, by Bragg’s law:

$$\begin{gathered} 2d\sin \theta =m\lambda \\ 2a\sin \varphi \times \sin \theta =m\lambda \\ 2a\sin \left(45°\right)\times \sin \left(40°\right)=m\lambda \\ 0.909a=\lambda \end{gathered}$$

For setup (a), we have . So, by Bragg’s law:

$$\begin{gathered} 2d\sin \theta =m\lambda \\ 2a\sin \theta =m\lambda \end{gathered}$$

Substitute :$\lambda =0.909a$

$$\begin{gathered} 2a\sin \theta =m\times 0.909a \\ \sin \theta =0.454m \end{gathered}$$

For , m =1 the value of$\theta =27°$ and m =2 for , the value of $\theta =65.4°$. We cannot take further values of because it will make the argument of the sine inverse function more than 1, which is not possible.