X-Ray Diffraction Allows information about the crystal structure of material

to be obtained.

(a) Destructive and (b) constructive interactions between x-rays and the crystal structure of a material. Diffraction occurs at angles that satisfy the Bragg equation.

X-rays that strike certain crystallographic planes at specific angles are reinforced and diffracted according to the Braggs law:

where = half the angle between the diffracted beam and the original beam direction, = wavelength of the x-raysdhkl = interplanar spacing between the planes that cause Bragg diffraction

hkld2sin =

222 lkhadhkl ++

=

(a) Schematic of an x-ray powder diffractometer showing the incident and diffracted beam, the sample in powdered form, and the movable x-ray detector. (b) The diffraction pattern of gold powder.

Analysis of the diffraction pattern allows the identification of the unknown material and the indexation of the diffracted peaks and their Bragg planes (hkl).

BCC: FCC:

)(4

sin 22222 lkh

a++=

)(4

sin 22222

2

lkha

++=

,......12,10,8,6,4,2222 =++ lkh

,......16,12,11,8,4,3222 =++ lkh

Example Problem

The results of an X-ray diffraction experiment using x-rays with = 0.07107 nm show that diffracted peaks occur at the following 2angles:

Peak 2 Peak 2

1 20.20 5 46.19

2 28.72 6 50.90

3 35.36 7 55.28

4 41.07 8 59.42

Determine the crystal structure, the indices of the plane producing each peak, and the lattice parameter (a) of the material.

2sin 222 lkh ++Peak 2 (hkl)1 20.20 0.0308 1 2 110

2 28.72 0.0615 2 4 200

3 35.36 0.0922 3 6 211

4 41.07 0.1230 4 8 220

5 46.19 0.1539 5 10 310

6 50.90 0.1847 6 12 222

7 55.28 0.2152 7 14 321

8 59.42 0.2456 8 16 400

0308.0sin2

So, the crystal structure of the material is BCC.

Use any diffracted peak to determine dhkl and , say peak 8 or (400)

So the material is BCC iron.

222400 lkhda ++=

nmd 071699.0)71.29sin(2

07107.0sin2400

===

= (0.071699)(4) = 0.2868 nm

nmaR 1242.04

3 ==

X-Ray DiffractionExample Problem