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CRYSTAL LATTICE

What is crystal (space) lattice? In

crystallography, only the geometrical properties

of the crystal are of interest, therefore one

replaces each atom by a geometrical point located

at the equilibrium position of that atom.

Platinum surface

Crystal lattice and structure of Platinum

Platinum

(scanning tunneling microscope)

Crystal Lattice

- An infinite array of points in space,
- Each point has identical surroundings to all

others. - Arrays are arranged exactly in a periodic manner.

Crystal Structure

- Crystal structure can be obtained by attaching

atoms, groups of atoms or molecules which are

called basis (motif) to the lattice sides of the

lattice point.

Crystal Structure Crystal Lattice Basis

A two-dimensional Bravais lattice with different

choices for the basis

Basis

- A group of atoms which describe crystal

structure

E

H

b) Crystal lattice obtained by identifying all

the atoms in (a)

a) Situation of atoms at the corners of regular

hexagons

Crystal structure

- Don't mix up atoms with lattice points
- Lattice points are infinitesimal points in space
- Lattice points do not necessarily lie at the

centre of atoms

Crystal Structure Crystal Lattice Basis

(No Transcript)

Types Of Crystal Lattices

- 1) Bravais lattice is an infinite array of

discrete points with an arrangement and

orientation that appears exactly the same, from

whichever of the points the array is viewed.

Lattice is invariant under a translation.

Types Of Crystal Lattices

2) Non-Bravais Lattice Not only the arrangement

but also the orientation must appear exactly the

same from every point in a bravais lattice.

- The red side has a neighbour to its immediate

left, the blue one instead has a neighbour to its

right. - Red (and blue) sides are equivalent and have the

same appearance - Red and blue sides are not equivalent. Same

appearance can be obtained rotating blue side

180º.

Translational Lattice Vectors 2D

- A space lattice is a set of points such that a

translation from any point in the lattice by a

vector - Rn n1 a n2 b
- locates an exactly equivalent point, i.e. a

point with the same environment as P . This is

translational symmetry. The vectors a, b are

known as lattice vectors and (n1, n2) is a pair

of integers whose values depend on the lattice

point.

P

Point D(n1, n2) (0,2) Point F (n1, n2)

(0,-1)

Lattice Vectors 2D

- The two vectors a and b form a set of lattice

vectors for the lattice. - The choice of lattice vectors is not unique. Thus

one could equally well take the vectors a and b

as a lattice vectors.

Lattice Vectors 3D

An ideal three dimensional crystal is described

by 3 fundamental translation vectors a, b and c.

If there is a lattice point represented by the

position vector r, there is then also a lattice

point represented by the position vector where u,

v and w are arbitrary integers.

r r u a v b w c (1)

Five Bravais Lattices in 2D

Unit Cell in 2D

- The smallest component of the crystal (group of

atoms, ions or molecules), which when stacked

together with pure translational repetition

reproduces the whole crystal.

2D-Crystal

S

S

Unit Cell

Unit Cell in 2D

- The smallest component of the crystal (group of

atoms, ions or molecules), which when stacked

together with pure translational repetition

reproduces the whole crystal.

2D-Crystal

The choice of unit cell is not unique.

b

a

2D Unit Cell example -(NaCl)

We define lattice points these are points with

identical environments

Choice of origin is arbitrary - lattice points

need not be atoms - but unit cell size should

always be the same.

This is also a unit cell - it doesnt matter if

you start from Na or Cl

- or if you dont start from an atom

This is NOT a unit cell even though they are all

the same - empty space is not allowed!

In 2D, this IS a unit cellIn 3D, it is NOT

Why can't the blue triangle be a unit cell?

Crystal Structure

22

Unit Cell in 3D

Crystal Structure

23

Unit Cell in 3D

Crystal Structure

24

Three common Unit Cell in 3D

Crystal Structure

25

Body centered cubic(bcc) Conventional ? Primitive

cell

Simple cubic(sc) Conventional Primitive cell

Crystal Structure

26

The Conventional Unit Cell

- A unit cell just fills space when translated

through a subset of Bravais lattice vectors. - The conventional unit cell is chosen to be larger

than the primitive cell, but with the full

symmetry of the Bravais lattice. - The size of the conventional cell is given by the

lattice constant a.

Crystal Structure

27

Primitive and conventional cells of FCC

Crystal Structure

28

Primitive and conventional cells of BCC

Primitive Translation Vectors

Primitive and conventional cells

Body centered cubic (bcc) conventional

¹primitive cell

Fractional coordinates of lattice points in

conventional cell 000,100, 010, 001, 110,101,

011, 111, ½ ½ ½

Simple cubic (sc) primitive cellconventional

cell

Fractional coordinates of lattice points 000,

100, 010, 001, 110,101, 011, 111

Crystal Structure

30

Primitive and conventional cells

Body centered cubic (bcc) primitive

(rombohedron) ¹conventional cell

Face centered cubic (fcc) conventional ¹

primitive cell

Fractional coordinates 000,100, 010, 001,

110,101, 011,111, ½ ½ 0, ½ 0 ½, 0 ½ ½ ,½1 ½ , 1 ½

½ , ½ ½ 1

Crystal Structure

31

Primitive and conventional cells-hcp

Hexagonal close packed cell (hcp) conventional

primitive cell

Fractional coordinates 100, 010, 110, 101,011,

111,000, 001

Crystal Structure

32

Unit Cell

- The unit cell and, consequently, the entire

lattice, is uniquely determined by the six

lattice constants a, b, c, a, ß and ?.

- Only 1/8 of each lattice point in a unit cell can

actually be assigned to that cell. - Each unit cell in the figure can be associated

with 8 x 1/8 1 lattice point.

Crystal Structure

33

Primitive Unit Cell and vectors

- A primitive unit cell is made of primitive

translation vectors a1 ,a2, and a3 such that

there is no cell of smaller volume that can be

used as a building block for crystal structures. - A primitive unit cell will fill space by

repetition of suitable crystal translation

vectors. This defined by the parallelpiped a1, a2

and a3. The volume of a primitive unit cell can

be found by - V a1.(a2 x a3) (vector products)

Cubic cell volume a3

Crystal Structure

34

Primitive Unit Cell

- The primitive unit cell must have only one

lattice point. - There can be different choices for lattice

vectors , but the volumes of these primitive

cells are all the same.

P Primitive Unit Cell NP Non-Primitive Unit

Cell

Crystal Structure

35

Wigner-Seitz Method

- A simply way to find the primitive
- cell which is called Wigner-Seitz
- cell can be done as follows
- Choose a lattice point.
- Draw lines to connect these lattice point to its

neighbours. - At the mid-point and normal to these lines draw

new lines. - The volume enclosed is called as a
- Wigner-Seitz cell.

Crystal Structure

36

Wigner-Seitz Cell - 3D

Crystal Structure

37

Lattice Sites in Cubic Unit Cell

Crystal Structure

38

Crystal Directions

- We choose one lattice point on the line as an

origin, say the point O. Choice of origin is

completely arbitrary, since every lattice point

is identical. - Then we choose the lattice vector joining O to

any point on the line, say point T. This vector

can be written as - R n1 a n2 b n3c
- To distinguish a lattice direction from a lattice

point, the triple is enclosed in square brackets

... is used.n1n2n3 - n1n2n3 is the smallest integer of the same

relative ratios.

Fig. Shows 111 direction

Crystal Structure

39

Examples

X ½ , Y ½ , Z 1 ½ ½ 1 1 1 2

Crystal Structure

40

Negative directions

- When we write the direction n1n2n3 depend on

the origin, negative directions can be written as

- R n1 a n2 b n3c
- Direction must be
- smallest integers.

Y direction

Crystal Structure

41

Examples of crystal directions

X -1 , Y -1 , Z 0 110

X 1 , Y 0 , Z 0 1 0 0

Crystal Structure

42

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