1. Miller Indices And X-ray diffraction
Chapter 3
Miller Indices
And
X-ray diffraction

2. Directions in a crystal lattice – Miller Indices
Vectors described by multiples of lattice constants: ua+vb+wc
e.g., the vector in the illustration crosses the edges of the unit cell at u=1, v=1, c=1/2
Arrange these in brackets, and clear the fractions:
[1 1 ½] = [2 2 1]

3. Negative directions have a bar over the number
Families of crystallographically equivalent directions, e.g., [100], [010], [001] are written as <uvw>, or, in this example, <100>

4. Directions in HCP crystals
a1, a2 and a3 axes are 120o apart, z axis is perpendicular to the a1,a2,a3 basal plane
Directions in this crystal system are derived by converting the [u′v′w′] directions to [uvtw] using the following convention:
n is a factor that reduces [uvtw] to smallest integers. For example, if
u′=1, v′=-1, w′=0, then
[uvtw]=

5. Crystallographic Planes
To find crystallographic planes are represented by (hkl). Identify where the plane intersects the a, b and c axes; in this case, a=1/2, b=1, c=∞
Write the reciprocals 1/a, 1/b, 1/c:
Clear fractions, and put into parentheses:
(hkl)=(210)

6. If the plane interesects the origin, simply translate the origin to an equivalent location.
Families of equivalent planes are denoted by braces:
e.g., the (100), (010), (001), etc. planes are denoted {100}

7. Planes in HCP crystals are numbered in the same way
e.g., the plane on the left intersects a1=1, a2=0, a3=-1, and z=1, thus the plane is

8. X-RAYS TO CONFIRM CRYSTAL STRUCTURE
• Incoming X-rays diffract from crystal planes.
Adapted from Fig. 3.2W, Callister 6e.
• Measurement of:
Critical angles, qc,
for X-rays provide
atomic spacing, d. 25

9. X-ray diffraction and crystal structure
X-rays have a wave length, l0.1-10Å.
This is on the size scale of the structures we wish to study
X-rays interfere constructively when the interplanar spacing is related to an integer number of wavelengths in accordance with Bragg’s law:

10. Because of the numbering system, atomic planes are perpendicular to their corresponding vector,
e.g., (111) is perpendicular to [111]
The interplanar spacing for a cubic crystal is:
Because the intensity of the diffracted beam varies depending upon the diffraction angle, knowing the angle and using Bragg’s law we can obtain the crystal structure and lattice parameter

11. Rules for diffracting planes
By comparing the ratios of the diffracted peaks, we can determine the ratios of the diffracting planes and determine the corresponding Miller indices

12. Bragg’s law only describes the size and shape of the unit cell If there are parallel planes inside the unit cell, their reflections can interfere constructively and result in zero intensity of the reflected beam hence, different crystal structures will only allow reflections of particular planes according to the following rules: