A beam of electrons strikes a crystal at an angle$\mathbf{\text{θ}}$with the atomic planes, reflects of many atomic planes below the surface, and then passes into a detector also making angle$\mathbf{\text{θ}}$with the atomic planes. (a) If the minimum$\mathbf{\text{θ}}$giving constructive interference is.$\mathbf{\text{35°}}$ What is the ratio$\mathbf{\text{λ/d}}$, Where is the spacing between atomic planes? (b) At what other angles, if any, would constructive interference occur?

(a)

The angle of constructive interference${\text{θ}}_{\text{min}}=\text{35}°$.

When there is constructive interference, Bragg's equation can be used to describe the relationship between the quantities.

$\mathbf{\text{2dsinθ=mλ………………(1)}}$

Using equation$\text{(1)}$.

$\begin{array}{l}\text{2dsinθ=mλ}\\ \text{2sinθ=}\frac{\text{mλ}}{\text{d}}\\ \frac{\text{λ}}{\text{d}}\text{=}\frac{\text{2sinθ}}{\text{m}}\end{array}$

${\text{θ}}_{\text{min}}$is the minimum angle for the constructive interference, so$\text{m=1}$.

$\begin{array}{c}\frac{\text{λ}}{\text{d}}\text{=}\frac{\text{2}\times \text{sinθ}}{\text{m}}\\ \text{=}\frac{\text{2}\times \text{sin}(\text{35}°)}{\text{1}}\\ \text{=1.15}\end{array}$

Therefore the ratio of$\frac{\text{λ}}{\text{d}}$ is $\text{1.15}$in the spacing between the atomic planes.

From Equation$\text{(1)}$the minimum value for$\text{sinθ}$is$\text{1}$.

So,

$\begin{array}{l}\text{2dsinθ=mλ}\\ \text{sinθ=}\frac{\text{mλ}}{\text{2d}}\\ \text{1}\ge \frac{\text{mλ}}{\text{2d}}\\ \text{m}\le \frac{\text{2}}{\frac{\text{λ}}{\text{d}}}\end{array}$

Insert$\frac{\text{λ}}{\text{d}}\text{=1.15}$

$\begin{array}{c}\text{m}\le \frac{\text{2}}{\frac{\text{λ}}{\text{d}}}\\ \le \frac{\text{2}}{\text{1.15}}\\ \text{=1.74}\end{array}$

Here m can notbe considered as $\text{2}$.So this is an invalid value of the.$\text{sinθ}$

As a result, the only integer that exists is$\text{m=1}$ .

This is the smallest angle at which constructive interference was detected. As a result, constructive interference is not visible from any other angles.