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
$n\lambda =2d\sin \theta \dots$ (i)
In case (1),
Given, $\lambda =3\mathring{A}$, $\theta =30^{\circ },n=2$
For $(100)$ plane $d_{hkl}=\frac{a}{\sqrt{(1{)}^2+(0{)}^2+(0{)}^2}}$
$\Rightarrow d=\frac{6}{\sqrt{1}}$
Substituting the above values in Eq. (i)
$2\times 3=2\times \frac{6}{\sqrt{1}}\sin 30\dots$(ii)
In case (2),
Given, $\lambda =3\mathring{A},n=1,d=6$
For $(200)$ plane
$d_{hkl}=\frac{a}{\sqrt{(2{)}^2+(0{)}^2+(0{)}^2}}$
$\Rightarrow d=\frac{6}{2}=3$
Substituting the above values in (1)
$1\times 3=2\times 3\sin \theta$
$\theta =30^{\circ }$