At what angles for the first order diffraction, spacing between two planes, respectively, are $\lambda $ and $\dfrac{\lambda }{2}$ ?
(A) ${0^ \circ },{90^ \circ }$
(B) ${90^ \circ },{0^ \circ }$
(C) ${30^ \circ },{90^ \circ }$
(D) ${90^ \circ },{30^ \circ }$

Hint: The equation that relates the interplanar distance and the angle of diffraction is the Bragg’s equation and it can be given as
\[2d\sin \theta = n\lambda \]

Complete step by step solution:
Bragg’s law gives the angles for the coherent and incoherent scattering of light from a crystal lattice. We know that in crystalline solid, the light waves are scattered from the lattice planes which are separated by the interplanar distance d.
- Scientist Bragg gave the relation between the path differences between the two waves undergo interference and diffraction angle. The Bragg’s equation is given as
\[2d\sin \theta = n\lambda \]
Where d is interplanar distance and n is a positive integer. $\lambda $ is the wavelength of the incident wave.
- We are provided with the question that the diffraction is of first order. So, the value of n is 1.
- Now, in one case, we are given that the interplanar distance is $\lambda $. So, in that case, the Bragg equation will be
\[2d\sin \theta = n\lambda \]
Putting the available values, we will get
\[2\lambda \sin \theta = (1)\lambda \]
So,
\[\sin \theta = \dfrac{\lambda }{{2\lambda }} = \dfrac{1}{2}\]
So, we can say that $\sin {30^ \circ } = \dfrac{1}{2}$ .
So, $\theta = {30^ \circ }$
In the second case, we are given that the interplanar distance is $\dfrac{\lambda }{2}$. So, putting this in the Bragg’s equation will give
\[2\left( {\dfrac{\lambda }{2}} \right)\sin \theta = (1)\lambda \]
So, we can write that
\[\sin \theta = \dfrac{{2\lambda }}{{2\lambda }} = 1\]
Thus, $\sin {90^ \circ } = 1$
So, we got that $\theta = {90^ \circ }$
Thus, we obtained that the angles will be ${30^ \circ }{\text{ and 9}}{{\text{0}}^ \circ }$ respectively.

Therefore, the correct answer is (C).

Note: Note that in the equation, n is the order of diffraction and it is always an integer value. With Bragg's law, we can find the lattice spacing for different cubic lattice systems which also includes the use of Miller indices.