Question

If the angle of incidence of X-ray of wavelength $3\mathring{A}$ which produces a second order diffracted beam from the $\left(100\right)$ planes in a simple cubic lattice with interlayer spacing $a=6\mathring{A}$ is ${30}^o$, the angle of incidence that produces a first-order diffracted beam from the ($200$) planes is:

• A. ${15}^o$
• B. ${45}^o$
• C. ${30}^o$
• D. ${60}^o$

Answer 1

Answer: (C)

${30}^o$

Given that, wavelength of X-rays $\left(\lambda \right)=3\mathring{A}$

Lattice constant $\left(a\right)=6\mathring{A}$

Plane $(h,k,l)=(2,0,0)$

Order of diffraction, $n=1$

Angle of incidence $\theta =?$

By, Bragg’s Diffraction law,

$2dsin\theta =n\lambda$....(i)

Now, $d=\frac{a}{\sqrt{h^2+k^2+l^2}}=\frac{6}{\sqrt{2^2+0^2+0^2}}=\frac{6}{\sqrt{4}}=\frac{6}{2}=3$

Putting values in equation (i)

Now, $2\times 3\times sin\theta =1\times 3$

$sin\theta =\frac{1}{2}$

$\theta ={30}^o$