91Ó°ÊÓ

Diffraction is a phenomena in which the interference or the bending of waves occurs around the corners of the opening or obstacle through or on which it strikes.

The diffraction formula is following,

$$\begin{gathered} \mathrm{²Ôλ}=2\mathrm{»å²õ¾\pm ²Ôθ} \\ \mathrm{λ}=\text{wavelength} \\ \mathrm{d}=\text{spacing} \\ \mathrm{θ}=\text{bragg angle} \\ \mathrm{n}=\text{order of diffraction} \end{gathered}$$

The shortest Wavelength${\mathrm{λ}}_1=430nm$

The longest Wavelength${\mathrm{λ}}_2=680nm$

The diffraction formula,

$$\begin{gathered} {\mathrm{λ}}_1=d\sin \mathrm{θ} \\ {\mathrm{λ}}_2=d\sin \left(\mathrm{θ}+\mathrm{Δ}\mathrm{θ}\right) \\ \sin \left(\mathrm{θ}+\mathrm{Δ}\mathrm{θ}\right)=\sin \mathrm{θ}\cos \mathrm{Δ}\mathrm{θ}+\cos \mathrm{θ}\sin \mathrm{Δ}\mathrm{θ} \end{gathered}$$

From above equation,

$sin\mathrm{θ}=\frac{{\mathrm{λ}}_1}{d}$

And,

$\cos \mathrm{θ}=\sqrt{1-{\left(\frac{{\mathrm{λ}}_1}{d}\right)}^2}$

After replacing the$\sin \mathrm{θ}$ and$\cos \mathrm{θ}$ in ${\mathrm{λ}}_2$equation,

${\mathrm{λ}}_1\cos \mathrm{Δ}\mathrm{θ}+\sqrt{d^2-{{\mathrm{λ}}_1}^2}\sin \mathrm{Δ}\mathrm{θ}={\mathrm{λ}}_2$

The value of d will be,

$d=\sqrt{\frac{{\left({\mathrm{λ}}_2-{\mathrm{λ}}_1\cos \mathrm{Δ}\mathrm{θ}\right)}^2+{\left({\mathrm{λ}}_1\sin \mathrm{Δ}\mathrm{θ}\right)}^2}{{\sin }^2\mathrm{Δ}\mathrm{θ}}}$

Substitute all the value in the above equation.

$$\begin{gathered} d=\sqrt{\frac{{\left(680 \text{nm}-430 \text{nm}×\cos 20°\right)}^2+{\left(430 \text{nm}\sin 20°\right)}^2}{{\sin }^{2}20°}} \\ d=914 \text{nm} \\ d=9.14×{10}^{-4} \text{mm} \end{gathered}$$

$$\begin{gathered} \frac{1}{d}=\frac{1}{9.14×{10}^{-4} \text{mm}} \\ \frac{1}{d}=1.09×{10}^3 \text{ruling per mm} \end{gathered}$$

Hence the number of ruling is, $1.09×{10}^3 \text{ruling per mm}$.