aula07-Fundamentos do Estado Cristalino

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Crystallographic Directions, and Planes
We need a way to identify directions and planes of atoms.
•Why?
Deformation under loading (slip) occurs on certain crystalline planes and in certain crystallographic directions. Before we can predict how materials fail, we need to know what modes of failure are more likely to occur.
Other properties of materials (electrical conductivity, thermal conductivity, elastic modulus) can vary in a crystal with orientation.
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• It is often necessary to be able to specify certain directions
and planes in crystals.
• Many material properties and processes vary with direction
in the crystal.
• Directions and planes are described using three integers -
Miller Indices.
Crystallographic direction is a vector [uvw]
Always passes thru origin 000
Measured in terms of unit cell dimensions a, b, and c
Smallest integer values
Planes with Miller Indices (hkl)
If plane passes thru origin, translate
Length of each planar intercept in terms of the lattice parameters a, b, and c.
Reciprocals are taken
If needed multiply by a common factor for integer representation
Crystallographic Directions
1. Vector repositioned (if necessary) to pass 
 through origin.
2. Read off projections in terms of 
 unit cell dimensions a, b, and c
3. Adjust to smallest integer values
4. Enclose in square brackets, no commas
	
	[uvw]
ex: 1, 0, ½
=> 2, 0, 1
=> [ 201 ]
-1, 1, 1
families of directions <uvw>
z
x
Algorithm
where overbar represents a negative index
[ 111 ]
=>
y
Indices of a Family or Form
 
Hexagonal Crystallographic Directions
1. Vector repositioned (if necessary) to pass 
 through origin.
2. Read off projections in terms of unit
 cell dimensions a1, a2, a3, or c
3. Adjust to smallest integer values
4. Enclose in square brackets, no commas
	
	[uvtw]
[ 1120 ]
ex: ½, ½, -1, 0 =>
Adapted from Fig. 3.8(a), Callister 7e.
dashed red lines indicate 
projections onto a1 and a2 axes
a1
a2
a3
-a3
2
a
2
2
a
1
-
a3
a1
a2
z
Algorithm
Hexagonal Crystals
4 parameter Miller-Bravais lattice coordinates are related to the direction indices (i.e., u'v'w') as follows.
=
=
=
'
w
w
t
v
u
)
v
u
(
+
-
)
'
u
'
v
2
(
3
n
-
)
'
v
'
u
2
(
3
n
-
=
]
uvtw
[
]
'
w
'
v
'
u
[
®
Fig. 3.8(a), Callister 7e.
-
a3
a1
a2
z
Hexagonal Crystallographic Directions
Crystallographic Planes
Crystallographic Planes
Miller Indices: Reciprocals of the (three) axial intercepts for a plane, cleared of fractions & common multiples. All parallel planes have same Miller indices.
Algorithm 
1.  Read off intercepts of plane with axes in 
 terms of a, b, c
2. Take reciprocals of intercepts
3. Reduce to smallest integer values
4. Enclose in parentheses, no 
 commas i.e., (hkl) 
Crystallographic Planes
z
x
y
a
b
c
4. Miller Indices (110)
example
a b c
z
x
y
a
b
c
4. Miller Indices (100)
1. Intercepts
1 1 
2. Reciprocals
1/1 1/1 1/
1 1 0
3. Reduction
1 1 0
1. Intercepts
1/2  
2. Reciprocals
1/½ 1/ 1/
2 0 0
3. Reduction
2 0 0
example
a b c
Crystallographic Planes
z
x
y
a
b
c
4. Miller Indices (634)
example
1. Intercepts
1/2 1 3/4
a b c
2. Reciprocals
1/½ 1/1 1/¾
2	1 4/3
3. Reduction
6	3 4
(001)
(010),
Family of Planes {hkl}
(100),
(010),
(001),
Ex: {100} = (100),
Crystallographic Planes (Hexagonal)
In hexagonal unit cells the same idea is used 
example
a1 a2 a3 c
4. Miller-Bravais Indices
(1011)
1. Intercepts
1 
-1
1
2. Reciprocals
1 1/
1 0 
-1
-1
1
1
3. Reduction
1 0
-1
1
a2
a3
a1
z
Adapted from Fig. 3.8(a), Callister 7e.
Interplanar Spacings
The inter-planar spacing (d) for a set of parallel planes will be important in crystallography. This value is related to the Miller indices and the unit cell dimensions.
For orthorhombic, tetragonal and cubic unit cells (the axes are all mutually perpendicular), the inter-planar spacing is given by:
as derived by geometry.
For other lattice types, use:
h, k, l = Miller indices
a, b, c = unit cell dimensions
Vetores da rede recíproca, ligada aos parâmetros da rede direta por
Esses vetores tem as seguintes propriedades 
De forma geral
Definamos agora o vetor em termos dos vetores da rede recíproca e dos índices de Miller como sendo 
Este vetor é perpendicular ao plano hkl, pois
O espaçamento interplanar é perpendicular aos planos
Assumindo que conheçamos os comprimentos dos parâmetros de rede é possível determinar 
Warren, B. E.; X-Ray Diffraction; Dover Publications INC, New York, USA, 1990
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