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