Lec. (4,5)

Z

X

Y

(100)

1

2

Crystal Systems

7 crystal systems

14 crystal lattices

Unit cell: smallest repetitive volume that contains the complete lattice pattern of a crystal.

a, b and c are the lattice constants

Unit Cells Types

Primitive Face-Centered

Body-Centered End-Centered

A unit cell is the smallest component of the crystal that reproduces the whole crystal when stacked together.

• Primitive (P) unit cells contain only a single lattice point.• Internal (I) unit cell contains an atom in the body center.• Face (F) unit cell contains atoms in the all faces of the planes composing the cell.• Centered (C) unit cell contains atoms centered on the sides of the unit cell.

Combining 7 Crystal Classes (cubic, tetragonal, orthorhombic, hexagonal, monoclinic, triclinic, trigonal) with 4 unit cell types (P, I, F, C) symmetry allows for only 14 types of 3-D lattice.

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(c) 2003 Brooks/Cole Publishing / Thomson Learning™

Lattice parameters in cubic, orthorhombic and hexagonal crystal systems.

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Define basic terms and give examples of each:

• Points (atomic positions)

• Vectors (defines a particular direction - plane normal)

• Miller Indices (defines a particular plane)

• relation to diffraction

• 3-index for cubic and 4-index notation for HCP

Basic definitions – Lattice sites

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Miller indices - A shorthand notation to describe certain crystallographic directions and planes in a material. Denoted by [ ], <>, ( ) brackets. A negative number is represented by a bar over the number.

Points, Directions and Planes in the Unit Cell

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• Coordinates of selected points in the unit cell.

• The number refers to the distance from the origin in terms of lattice parameters.

Point Coordinates

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Point CoordinatesPoint coordinates for unit cell

center are

a/2, b/2, c/2 ½½ ½

Point coordinates for unit cell corner are 111

Translation: integer multiple of lattice constants identical position in another unit cell

z

x

y

a b

c

000

111

y

z

2c

b

b

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Crystallographic Directions

1. Vector repositioned (if necessary) to pass through origin.2. Read off projections in terms of

unit cell dimensions a, b, and c3. Adjust to smallest integer values4. Enclose in square brackets, no commas

[uvw]

ex: 1, 0, ½ => 2, 0, 1 => [ 201 ]

-1, 1, 1

z

x

where overbar represents a negative index

[ 111 ]=>

y

[ 111 ]

[ 201 ]

Algorithm

Procedure:

1. Any line (or vector direction) is specified by 2 points.• The first point is, typically, at the origin (000).

2. Determine length of vector projection in each of 3 axes in

units (or fractions) of a, b, and c.• X (a), Y(b), Z(c)

1 1 0

3. Multiply or divide by a common factor to reduce the

lengths to the smallest integer values, u v w.

4. Enclose in square brackets: [u v w]: [110] direction.

a b

c

DIRECTIONS will help define PLANES (Miller Indices or plane normal).

[ 1 1 0]5. Designate negative numbers by a bar

• Pronounced “bar 1”, “bar 1”, “zero” direction.

6. “Family” of [110] directions is designated as <110>.

Directions in a Crystal

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Examples

210

X = ½ , Y = ½ , Z = 1

[½ ½ 1] [1 1 2]

X = 1 , Y = ½ , Z = 0

[1 ½ 0] [2 1 0]

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• When we write the direction

[n1n2n3] depending on the origin,

negative directions are written as

R = n1a1 + n2a2 + n3a3

To specify the direction, the

smallest possible integers must

be used.

Y direction

(origin) O

- Y direction

X direction

- X direction

Z direction

- Z direction

][ 321 nnn

Negative Directions

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X = -1 , Y = -1 , Z = 0 [110]X = 1 , Y = 0 , Z = 0 [1 0 0]

Examples of Crystal Directions

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X =-1 , Y = 1 , Z = -1/6

[-1 1 -1/6] [6 6 1]

A vector can be moved to the origin.

Examples

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• Within a crystal lattice it is possible to identify sets of

equally spaced parallel planes.

These are called lattice planes.

• In the figure, the density of lattice points on each plane of a set

is the same & all lattice points are contained on each set of

planes.

b

a

b

a

The set of

planes for a

2D lattice.

Crystal Planes

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• If the plane passes through origin, either:

– Construct another plane, or

– Create a new origin

– Then, for each axis, decide whether plane intersects or parallels the axis.

• Algorithm for Miller indices1. 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.16

Crystallographic Planes

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Crystallographic Planes• Crystallographic planes are specified by 3 Miller

Indices (h k l). All parallel planes have same Miller indices.

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• Miller Indices are a symbolic vector representation for the orientation of an atomic plane in a crystal lattice & are defined as the reciprocals of the fractional intercepts which the plane makes with the crystallographic axes.

• To find the Miller indices of a plane, take the following steps:

1. Determine the intercepts of the plane along each of the three crystallographic directions.

2. Take the reciprocals of the intercepts.

3. If fractions result, multiply each by the denominator of the smallest fraction.

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Crystallographic Planesz

x

y

a b

c

4. Miller Indices (110)

example a b cz

x

y

a b

c

4. Miller Indices (200)

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 0

3. Reduction 2 0 0

example a b c

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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

x y z

Intercepts

Intercept in terms of lattice parameters

Reciprocals

Reductions

Enclosure

a -b c/2

-1 1/20 -1 2

N/A

(012)

Determine the Miller indices for the plane shown in the sketch

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Family of Planes

• Planes that are crystallographically

equivalent have the same atomic packing.

• Also, in cubic systems only, planes having

the same indices, regardless of order and

sign, are equivalent.

• Ex: {111}

= (111), (111), (111), (111), (111), (111), (111), (111)

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

_ __ __ _ __ _ _ _ _

Family of planes{110}

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FCC Unit Cell with (110) plane

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BCC Unit Cell with (110) plane

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Axis X Y Z

Intercept

points 1 ∞ ∞

Reciprocals 1/1 1/ ∞ 1/ ∞Smallest

Ratio 1 0 0

Miller İndices (100)

Example-1

(1,0,0)

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Axis X Y Z

Intercept

points 1 1 ∞

Reciprocals 1/1 1/ 1 1/ ∞Smallest

Ratio 1 1 0

Miller İndices (110)

Example-2

(1,0,0)

(0,1,0)

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Axis X Y Z

Intercept

points 1 1 1

Reciprocals 1/1 1/ 1 1/ 1Smallest

Ratio 1 1 1

Miller İndices (111)(1,0,0)

(0,1,0)

(0,0,1)

Example-3

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Axis X Y Z

Intercept

points 1/2 1 ∞

Reciprocals 1/(½) 1/ 1 1/ ∞Smallest

Ratio 2 1 0

Miller İndices (210)(1/2, 0, 0)

(0,1,0)

Example-4

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Miller Indices

Reciprocal numbers are: 2

1 ,

2

1 ,

3

1

Plane intercepts axes at cba 2 ,2 ,3

Indices of the plane (Miller): (2,3,3)

(200) (100)

Indices of the direction: [2,3,3]a

3

2

2

bc

[2,3,3]

Z

X

Y

(100)

Z

X

Y

(110)

Z

X

Y

(111)

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Example 7

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SUMMARY• Crystallographic points, directions and planes are

specified in terms of indexing schemes.

• Materials can be single crystals or polycrystalline.

• Material properties generally vary with single crystal orientation (anisotropic), but are generally non-directional (isotropic) in polycrystals with randomly oriented grains.

• Some materials can have more than one crystal structure. This is referred to as polymorphism (or allotropy).

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A crystal resides in real space. The diffraction pattern of the crystal in Fraunhofer

diffraction geometry resides in Reciprocal Space. In a diffraction experiment

(powder diffraction using X-rays, selected area diffraction in a TEM), a part of

this reciprocal space is usually sampled.

From the real lattice the reciprocal lattice can be geometrically constructed. The

properties of the reciprocal lattice are ‘inverse’ of the real lattice → planes ‘far

away’ in the real crystal are closer to the origin in the reciprocal lattice.

As a real crystal can be thought of as decoration of a lattice with motif; a

reciprocal crystal can be visualized as a Reciprocal Lattice decorated with a

motif* of Intensities.

Reciprocal Crystal = Reciprocal Lattice + Intensities as Motif*

The reciprocal of the ‘reciprocal lattice’ is nothing but the real lattice!

Planes in real lattice become points in reciprocal lattice and vice-versa.

Reciprocal Lattice

In diffraction patterns (Fraunhofer geometry) (e.g. SAD), planes are mapped as spots (ideally points).

Hence, we would like to have a construction which maps planes in a real crystal as points.

Apart from the use in ‘diffraction studies’ we will see that it makes sense to use reciprocal lattice

when we are dealing with planes.

Motivation for constructing reciprocal lattices

As the index of the plane increases

→ the interplanar spacing decreases

→ and ‘planes start to crowd’ around

the origin in the real lattice (refer figure).

Hence, we work in reciprocal lattice

when dealing with planes.

As seen in the figure the diagonal is divided

into (h + k) parts.

Let us start with a one dimensional lattice and construct the reciprocal lattice

Reciprocal Lattice

Real Lattice

How is this reciprocal lattice constructed

Note in 1D planes are points and

have Miller indices of single digit

(they have been extended into the

second dimension (as lines) for

better visibility

The plane (2) has intercept at ½, plane (3) has intercept at 1/3 etc.

As the index of the plane increases it gets closer to the origin (there is

a crowding towards the origin)

One unit cell

Reciprocal Lattice

Each one of these points

correspond to a set of

‘planes’ in real space

Note that in reciprocal space index

has NO brackets

Real Lattice

Reciprocal Lattice

(01)

(10)(11)

(21)

10 20

11

221202

01 21

00

The reciprocal lattice has an origin!

1a

2a

1a1

1a

*

11g *

21g*b2

*b1

Now let us construct some 2D reciprocal lattices

Example-1

Each one of these points correspond to

a set of ‘planes’ in real space

2

1a

Note that vectors in reciprocal

space are perpendicular to planes

in real space (as constructed!)

But do not measure distances from the figure!

Overlay of real and

reciprocal lattices

g vectors connect

origin to reciprocal

lattice points