Question

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

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

Answer 1

Answer: (C)

According to Bragg's equation
$n \lambda=2 d \sin \theta \ldots$ (i)
In case (1),
Given, $\lambda=3 \, \mathring{A}$, $\theta=30^{\circ}, n=2$
For $(100)$ plane $d_{h k l}=\frac{a}{\sqrt{(1)^{2}+(0)^{2}+(0)^{2}}}$
$\Rightarrow d=\frac{6}{\sqrt{1}}$
Substituting the above values in Eq. (i)
$2 \times 3=2 \times \frac{6}{\sqrt{1}} \sin 30 \ldots$(ii)
In case (2),
Given, $\lambda=3 \, \mathring{A}, n=1, d=6$
For $(200)$ plane
$d_{h k l} =\frac{a}{\sqrt{(2)^{2}+(0)^{2}+(0)^{2}}} $
$\Rightarrow d =\frac{6}{2}=3$
Substituting the above values in (1)
$1 \times 3 =2 \times 3 \sin \theta $
$\theta =30^{\circ}$