Lecture 6Crystal Geometry

and Structure

Determination

Miller Indices of directions and planes

William Hallowes Miller(1801 – 1880)

University of Cambridge

1. Choose a point on the direction as the origin.2. Choose a coordinate system with axes parallel to the unit cell edges.

x

y3. Find the coordinates of another point on the direction in terms of a, b and c

4. Reduce the coordinates to smallest integers.

5. Put in square brackets

Miller Indices of Directions

[100]

1a+0b+0c

z

1, 0, 0

1, 0, 0

Note that we use separator only when one of them in double digits

y

z

All parallel directions pointing in same direction will have the same Miller indices

[100]x

x

y

z

O

A 1/2, 1/2, 1

[1 1 2]

OA=1/2 a + 1/2 b + 1 c

P

Q

x

y

z

PQ = -1 a -1 b + 1 c

-1, -1, 1

Miller Indices of Directions (contd.)

[ 1 1 1 ]__

-ve steps are shown as bar over the number

Some important directions in cubic lattice

Draw [221] direction

Family of Directions

Direction which looks physically identical but not parallelto some other direction comes under family of directions

A set of directions related by symmetry operationsof the lattice or the crystal is called a family of directions

For instance properties measured along these directions would be same owing to the symmetry of the crystal

Take an example of Cubic-F, face diagonals

All the equivalent directions can be represented by <u v w>

Miller indices of a family of symmetry related directions

[100]

[001]

[010]

uvw = [uvw] and all other directions related to [uvw] by the symmetry of the crystal

cubic100 = [100], [010], [001]

tetragonal100 = [100], [010]

Cubic Tetragonal

[010]

[100]

Family of directions

IndexNumber in the family for cubic

lattice

<100> → 3 x 2 = 6

<110> → 6 x 2 = 12

<111> → 4 x 2 = 8

These directions are very important for cubic latticewhen we will talk about properties

Negatives (opposite directions)

Miller Indices for planes

5. Enclose in parenthesis

Miller Indices for planes

3. Take reciprocal

2. Find intercepts along axes in

terms of respective lattice

parameters

1. Select a crystallographic

coordinate system with origin not

on the plane

4. Convert to smallest integers in

the same ratio

1 1 1

1 1 1

1 1 1

(111)

x

y

z

O

Octahedral plane

Again note that there is no separator

Miller Indices for planes (contd.)

origin

intercepts

reciprocals

Miller Indices

AB

CD

O

ABCD

O

1 ∞ ∞

1 0 0

(1 0 0)

OCBE

O*

1 -1 ∞

1 -1 0

(1 1 0)_

Plane

x

z

y

O*

x

z

E

Zero represents

that the plane is parallel to

the corresponding

axis

Bar represents a negative intercept

Crystallographically equivalent planes:Family of planes

All members physically identical butnot parallel to one and other.

Planes related with symmetry operation

Miller indices of a family of symmetry related planes

= (hkl ) and all other planes related to (hkl ) by the symmetry of the crystal

{hkl }

All the faces of the cube are equivalent to each other by symmetry

Front & back faces: (100)

Left and right faces: (010)

Top and bottom faces: (001)

{100} = (100), (010), (001)

{100}cubic = (100), (010), (001)

{100}tetragonal = (100), (010)

(001)

Cubic

Tetragonal

Miller indices of a family of symmetry related planes

x

z

y

z

x

y

CUBIC CRYSTALS

[hkl] (hkl)

Angle between two directions [h1k1l1] and [h2k2l2]:

C

[111]

(111)

22

22

22

21

21

21

212121coslkhlkh

llkkhh

Weiss zone law

True for ALL crystal systems

Not in the textbook

• If a direction [uvw] lies in a plane (hkl) then

• uh+vk+wl = 0

(hkl)

dhklInterplanar spacing between ‘successive’ (hkl) planes passing through the corners of the unit cell

222 lkh

acubichkld

O

x(100)

ad 100

B

O

x

z

E

2011

ad

INTERPLANAR SPACING

[uvw] Miller indices of a direction (i.e. a set of parallel directions)

(hkl) Miller Indices of a plane (i.e. a set of parallel planes)

<uvw> Miller indices of a family of symmetry related directions

{hkl} Miller indices of a family of symmetry related planes

No separators are allowed in MI of directions and planes

Unless the magnitude is in double digit

Summary of Notation convention for Indices

Structure Determination

How do we determine the structure of a piece of crystalline solid?

We often say something is Cubic-F….

You can probe the atomic arrangements by X-ray diffraction (XRD)

How Characteristic X-rays are generated??

25

Characteristic X-rays are produced by electron transitionsbetween the electron shells.

X-rays: Characteristic Radiation, K

Target (Anode)

Mo

Cu

Co

Fe

Cr

Wavelength, Å

0.71

1.54

1.79

1.94

2.29

Note that wavelength is typically ranging between 1-2 Å