3606 X-ray Diffraction

A crystal serves as a three-dimensional diffraction grating for x rays with wavelengths of the same order of magnitude as the spacing between atoms in the crystal. For a set of crystal planes spaced a distance d apart, constructive interference occurs when the angles of incidence and scattering (measured from the crystal planes) are equal and when the Bragg condition [Eq. (36.16)] is satisfied. (See Example 36.5.)

Bragg condition for constructive interference from an array:

$2d\sin \th = m \lambda \quad (m=1, 2, \cdots) \quad \bold{(36.16)}$

$d$ - Distance between adjacent rows in array
$\th$ - Angle of line from surface of array to mth bright region on screen

Exercises

34, 35

36.34 If the planes of a crystal are 3.50 $\text{\AA}$ ($1 \text{ \AA} = 10^{-10} \text{ m} = 1 \text{ \AA ngstrom unit}$) apart,
(a) what wavelength of electromagnetic waves is needed so that the first strong interference maximum in the Bragg reflection occurs when the waves strike the planes at an angle of 22.0$\degree$, and in what part of the electromagnetic spectrum do these waves lie? (See Fig. 32.4.)
(b) At what other angles will strong interference maxima occur?

36.35 X rays of wavelength 0.0850 nm are scattered from the atoms of a crystal. The second-order maximum in the Bragg reflection occurs when the angle $\th$ in Fig. 36.22 is 21.5$\degree$. What is the spacing between adjacent atomic planes in the crystal?