Chapter 3

Miller Indices And

X-ray diffraction

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]

Negative directions have a bar over the number

e.g., _

11

Families of crystallographically equivalent directions, e.g., [100], [010], [001] are written as <uvw>, or, in this example, <100>

Directions in HCP crystals

'

)''(

)''2(3

)''2(3

][]'''[

nww

vut

uvn

v

vun

u

uvtwwvu

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]= ]0011[_

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:

11

1211

l

k

h

Clear fractions, and put into parentheses:

(hkl)=(210)

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}

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 )1110(

_

d=n/2sinc

x-ray intensity (from detector)

c25

• Incoming X-rays diffract from crystal planes.

• Measurement of: Critical angles, c, for X-rays provide atomic spacing, d.

Adapted from Fig. 3.2W, Callister 6e.

X-RAYS TO CONFIRM CRYSTAL STRUCTURE

reflections must be in phase to detect signal

spacing between planes

d

incoming

X-rays

outg

oing

X-ra

ys

detector

extra distance travelled by wave “2”

“1”

“2”

“1”

“2”

X-ray diffraction and crystal structure

• X-rays have a wave length, Å.• 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:

sin2dn

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:

222 lkh

adhkl

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

Rules for diffracting planes

2211 sinsin dd By comparing the ratios of the diffracted peaks, we can determine the ratios of the diffracting planes and determine the corresponding Miller indices

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 beamhence, different crystal structures will only allow reflections of particular planes according to the following rules: