Chapter 25, Problem 56P

To determine

Whether both wavelength and lattice spacing can be determined by measuring the diffraction peaks corresponding to the first three orders.

Answer to Problem 56P

Solution:

By measuring the diffraction peaks corresponding to the first three orders alone, both wavelength and lattice spacing cannot be determined.

Explanation of Solution

Bragg’s law gives the expression that relates the lattice spacing, wavelength and the glancing angle.

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,

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

The peaks correspond to maximum intensity and the glancing angle corresponding to the peaks can be measured.

For the first, second and the third orders, the Bragg’s law expression is written as,

$\begin{array}{c}\lambda =2d\sin {\varphi }_1\\ 2\lambda =2d\sin {\varphi }_2\\ 3\lambda =2d\sin {\varphi }_3\end{array}$

The three equations given above can merely provide relationships between the glancing angles, but can return any values of $\left(\lambda ,d\right)$. To determine the lattice spacing, the value of the wavelength must be known and vice a versa.