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STRUCTURE OF MATERIALSCh07: Crystallographic PlanesPROF. DR. RAMIS MUSTAFA ÖKSÜZOĞLU
DR. UMUT SAVACI
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CRYSTALLOGRAPHIC NOTATION: MILLER INDICESMiller indices form a notation system in crystallography forplanes in crystal (Bravais) lattices.
In particular, a family of lattice planes is determined bythe Miller indices. They are written (hkℓ), and each indexdenotes a plane orthogonal to a direction [h k ℓ] inthe basis of the reciprocal lattice vectors.
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Crystallographic Planes
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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.
• Algorithm1. Read off intercepts of plane with axes in
terms of a, b, c2. Take reciprocals of intercepts3. Reduce to smallest integer values4. Enclose in parentheses, no commas i.e., (hkl)
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Crystallographic Planesz
x
ya b
c
4. Miller Indices (110)
example a b cz
x
ya b
c
4. Miller Indices (100)
1. Intercepts 1 1
2. Reciprocals 1/1 1/1 1/
1 1 03. Reduction 1 1 0
1. Intercepts 1/2
2. Reciprocals 1/½ 1/ 1/
2 0 03. Reduction 2 0 0
example a b c
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Crystallographic Planes
z
x
ya b
c
4. Miller Indices (634)
example1. 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
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https://www.doitpoms.ac.uk/tlplib/miller_indices/lattice_draw.php
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Equivalent Planes
Note the shift of origin from blue to red circle for the negative indices
(001)
(010),
Family of Planes {hkl}
(100), (010),
(001),{100} = (100),
{110} = HOMEWORK
{111} = HOMEWORK
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Crystallographic Planes (HCP)
example a1 a2 a3 c
4. Miller-Bravais Indices (1011)
1. Intercepts 1 -1 12. Reciprocals 1 1/
1 0
-1
-1
1
1
3. Reduction 1 0 -1 1
a2
a3
a1
z
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Crystallographic Planes (HCP)
11 21 10 11
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Planar Density of (100) Iron
(100)
Radius of iron R = 0.1241 nm
R3
34a =
Adapted from Fig. 3.2(c), Callister 7e.
2D repeat unit
= Planar Density =a2
1
atoms
2D repeat unit
= nm2
atoms12.1
m2
atoms= 1.2 x 1019
1
2
R3
34area
2D repeat unit
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Interplanar SpacingThe spacing between planes in a crystal is known as interplanar spacing and is denoted as dhkl
)In cubic system
hkl
d111 = a/ 3
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X-Rays to Determine Crystal Structure
X-ray intensity (from detector)
q
qc
d =nl
2 sinqc
Measurement of
critical angle, qc,
allows computation of
planar spacing, d.
Adapted from Fig. 3.19,
Callister 7e.
reflections must be in phase for a detectable signal
spacing between planes
d
ql
q
extra distance travelled by wave “2”
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X-Ray Diffraction Pattern
(110)
(200)
(211)
z
x
ya b
c
Diffraction angle 2q
Diffraction pattern for polycrystalline a-iron (BCC)
Inte
nsity (
rela
tive)
z
x
ya b
c
z
x
ya b
c
d =nl
2 sinqc