Question

X-rays of wavelength $\mathbf{2}\mathbf{.}\mathbf{63}\mathbf{}\mathbf{\AA }$were used to analyze a crystal. The angle of first-order diffraction (n=1 in the Bragg equation) was $\mathbf{15}\mathbf{.}\mathbf{55}\mathbf{°}$. What is the spacing between crystal planes, and what would be the angle for second-order diffraction (n=2)?

Short answer

The spacing between crystal planes is $\mathrm{d}=4.91\times {10}^{-10}\mathrm{m}$and the angle for second-order diffraction is ${\mathrm{\theta }}_2=32.4°$.

Step-by-step solution

Given Information

The wavelength of the x-ray is $\mathrm{\lambda }=2.63\mathrm{\AA }\left(\frac{{10}^{-10}\mathrm{m}}{1\mathrm{\AA }}\right)=2.63\times {10}^{-10}\mathrm{m}$

The angle for the first order diffraction (n=1) is${\mathrm{\theta }}_1=15.55°$

Concept Introduction

According to Bragg’s equation,

$\mathbf{n\lambda }\mathbf{=}\mathbf{2}\mathbf{dsin\theta }$

Where n is the order of diffraction, d is the distance between the atomic layers and $\mathbf{\lambda }$is the wavelength of the incident X-ray.

Calculation of the spacing between the crystal planes d

Rearranging Bragg's equation in terms of d, we get,

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

Now, substitute the given data for first order diffraction, and we get,

localid="1663780045492" $$\begin{gathered} \mathrm{d}=\frac{1\times 2.63\times {10}^{-10}\mathrm{m}}{2\times \sin \left(15.55°\right)} \\ \mathrm{d}=4.91\times {10}^{-10}\mathrm{m} \end{gathered}$$

Thus the interplanar spacing is localid="1663780079509" $\mathrm{d}=4.91\times {10}^{-10}\mathrm{m}$.

Now use this value to calculate the diffraction angle for second order diffraction (n=2), so, substitute the values in Bragg's equation, which gives,

$$\begin{gathered} 2\times 2.63\times {10}^{-10}\mathrm{m}=2\times 4.91\times {10}^{-10}\mathrm{m}\times {\mathrm{sin\theta }}_2 \\ \sin {\mathrm{\theta }}_2=\frac{2\times 2.63\times {10}^{-10}\mathrm{m}}{2\times 4.91\times {10}^{-10}\mathrm{m}} \\ \sin {\mathrm{\theta }}_2=0.5356 \\ {\mathrm{\theta }}_2={\sin }^{-1}\left(0.5356\right) \\ {\mathrm{\theta }}_2=32.4° \end{gathered}$$

Thus the diffraction angle is${\mathrm{\theta }}_2=32.4°$