STRUCTURE OF MATERIALS Ch07: Crystallographic Planes PROF. DR. RAMIS MUSTAFA ÖKSÜZOĞLU DR. UMUT SAVACI
STRUCTURE OF MATERIALSCh07: Crystallographic PlanesPROF. DR. RAMIS MUSTAFA ÖKSÜZOĞLU
DR. UMUT SAVACI
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.
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.
• 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)
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
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
https://www.doitpoms.ac.uk/tlplib/miller_indices/lattice_draw.php
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
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
Crystallographic Planes (HCP)
11 21 10 11
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
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
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”
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