Chapter 25, Problem 54P

To determine

The angle relative to the surface, the X rays must be directed to observe the first diffraction maximum is to be observed.

Answer to Problem 54P

Solution:

To observe the first diffraction maximum, the X rays must be directed at an angle 14^o relative to the surface.

Explanation of Solution

X rays are electromagnetic waves of very short wavelengths. When X rays are directed on to the surface of a crystal, they are diffracted since the crystal behaves as a three dimensional diffraction grating. The diffracted beams produce maxima and minima depending on their path difference.

If the spacing between the planes of a crystal is d and the X rays of wavelength $\lambda$ are incident on the planes at a glancing angle $\varphi$, then according to Bragg’s equation, the m^th maximum is obtained according to the expression,

$m\lambda =2d\sin \varphi$

Given:

The wavelength of the X rays $\lambda =0.138\text{ nm}$

The distance between the planes $d=0.285\text{ nm}$

Order of diffraction $m=1$

Formula:

The glancing angle is calculated using the expression,

$m\lambda =2d\sin \varphi$

Calculation:

Substitute the given values of wavelength, plane separation and the order of diffraction in the expression and evaluate.

$\begin{array}{c}m\lambda =2d\sin \varphi \\ \sin \varphi =\frac{m\lambda }{2d}\\ =\frac{1\left(0.138\text{ nm}\right)}{2\left(0.285\text{ nm}\right)}\\ =0.2421\end{array}$

Take the inverse of the sine function to get the angle.

$\begin{array}{c}\varphi ={\sin }^{-1}\left(0.2421\right)\\ =14°\end{array}$