Transcript Chapter 3 Miller Indices And X-ray diffraction

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.,
1  1
Families of
crystallographically
equivalent directions, e.g.,
[100], [010], [001] are
written as <uvw>, or, in
this example, <100>
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:
[u ' v' w' ]  [uvtw]
n
u  (2u 'v' )
3
n
v  (2v'u ' )
3
t  (u ' v' )
w  nw'
n is a factor that reduces
[uvtw] to smallest integers.
For example, if
u′=1, v′=-1, w′=0, then
_
[uvtw]= [11 00]
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:
1
1
2
1
k
1
1
l

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 (10 11)
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.
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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:
nl  2d sin q
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:
d hkl 
a
h2  k 2  l 2
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
l  d1 sin q1  d2 sin q2
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 beam
hence, different crystal structures will only allow reflections of particular
planes according to the following rules: