PRESENTATION

ON

SPACE LATTICE AND

CRYSTAL STRUCTURE

Submitted By

Manmeet Singh

13209015

ME Mechanical

Submitted To

Dr. Sanjeev Kumar

Mechanical Deptt.

MATERIALS SOLIDS

1. Crystalline

a. Single crystalline

b. Poly crystalline

“High Bond Energy” and a More Closely Packed Structure

2. Non Crystalline(Amorphous)

These have less densely packed lower bond energy

“structures”

• Non dense, random packing

• Dense, ordered packing

Dense, ordered packed structures tend to have

lower energies & thus are more stable.

ENERGY AND PACKING

Energy

r

typical neighbor bond length

typical neighbor bond energy

Energy

r

typical neighbor bond length

typical neighbor bond energy

CRYSTALS GEOMETRY

Space lattice

Crystal structure

Crystal directions and planes

SPACE LATTICE

Arrangement of atoms taken as point periodically

repeating in infinite array in 3 dimensional space

such that every point has identical surrounding.

Infinite

Identical surrounding

In a space lattice we can have more than one kind of

cell, shape of cell, size of cell…..

Meaning of

identical

surrounding

TO DEFINE MOVEMENT

2d – two vectors which are non collinear

3d – three vectors which are non coplanar

PRIMITIVE CELL SMALLEST CELL WITH LATTICE POINTS AT EIGHT

CORNERS HAS EFFECTIVELY ONLY ONE ATOM IN THE

VOLUME OF CELL

PARAMETERS OF PRIMITIVE CELL

3D lattices can be generated with three basis vectors

6 lattice parameters

3 distances (a, b, c)

3 angles (, , )

PARAMETERS OF PRIMITIVE CELL

a,b,c adjacent

sides of cell

α,β,γ interfacial

angles

Points to note----

1. Knowing actual values of (a,b,c) and (α,β,γ) ---form and size of cell.

2. Knowing (α,β,γ) but only rations of (a,b,c)---we can only know

shape of cell not size.

UNIT CELL NOT NECESSARY TO BE PRIMITIVE, CAN BE BIGGER THAN

PRIMITIVE AS LONG AS IT SHOWS ALL POSSIBLE MAXIMUM

SYMMETRIES

It is characterized by:

Type of atom and their radii,

Cell dimensions,

Number of atoms per unit cell,

Coordination number (CN)–closest neighbors to an atom

Atomic packing factor, APF

Atomic packing factor (APF) or packing fraction is the fraction of volume in a crystal structure that is occupied by atoms.

CHOOSING UNIT CELL--- BRAVAIS SPACE LATTICES

Smallest size

Maximum possible symmetry

Symmetry

1. Translational symmetry (inherent in definition

of space lattice is that identical surrounding)

2. Rotation symmetry

So there are only 14 bravais space lattices

which belong to 7 crystal classes.

Depending on minimum size and maximum

symmetry.

CRYSTAL STRUCTURE

Crystal– considered as consisting of tiny blocks which are repeated in 3D pattern.

Tiny block--- UNIT CELL

Unit cell---arrangement of small group of atoms . It is that volume of solid from which entire crystal can be constructed by repeated translation in 3D.

Lattice point --- each atom in cell is replaced by point.

Space lattice(infinite lattice array) --- arrangement of lattice in 3D.every point has identical surrounding.

Lattice spacing--- distance between atom points.

DEFINING CRYSTAL STRUCTURE

A crystal structure is a unique arrangement of atoms or molecules in a crystalline solid. A crystal structure describes a highly ordered structure, occurring due to the intrinsic nature of molecules to form symmetric patterns.

Inter atomic spacing

Number of atoms and their kind

Orientation in space

MOTIF –kind of atoms associated with each lattice point

in polymeric and protein crystals there can be more than 10,000 atoms in a motive

ARRANGEMENT OF LATTICE POINTS IN

SPACE LATTICE

1. Primitive simple cubic

8 points or atoms at

each corner

2. 8 corner atom and one

body centered

3. 8 atoms at each

corner and 6 atoms at

center of each face.

4. 8 atoms at each

corner and 2 atoms at

centers of opposite

face.

4. Hexagonal with 12

corner atoms and 2

center atoms

7 CRYSTAL CLASSES

FIRST CRYSTAL CLASS CUBIC CRYSTAL CLASS

Symmetry --- THREE -4 fold

1. Simple cubic

2. Body centered cubic

3. Face centered cubic

a = b= c

α = β = γ = 90°

Only one parameter need to be defined

No end center as they do not show maximum symmetry

• Rare due to low packing density (only Po – Polonium

-- has this structure)

• Coordination No. =

6

(# nearest neighbors)

for each atom as seen

SIMPLE CUBIC STRUCTURE (SC)

• APF for a simple cubic structure = 0.52

APF =

a 3

4

3 p (0.5a) 3 1

atoms

unit cell atom

volume

unit cell

volume

ATOMIC PACKING FACTOR (APF)

APF = Volume of atoms in unit cell*

Volume of unit cell

*assume hard spheres

close-packed directions

a

R=0.5a

contains (8 x 1/8) = 1 atom/unit cell Here: a = Rat*2

Where Rat’ atomic radius

• Coordination # = 8

• Atoms touch each other along cube diagonals within

a unit cell.

--Note: All atoms are identical; the center atom is shaded

differently only for ease of viewing.

BODY CENTERED CUBIC STRUCTURE (BCC)

ex: Cr, W, Fe (), Tantalum, Molybdenum

2 atoms/unit cell: (1 center) + (8 corners x 1/8)

ATOMIC PACKING FACTOR: BCC

a

APF =

4

3 p ( 3 a/4 ) 3 2

atoms

unit cell atom

volume

a 3

unit cell

volume

length = 4R =

Close-packed directions:

3 a

• APF for a body-centered cubic structure = 0.68

a R

a 2

a 3

• Coordination # = 12

• Atoms touch each other along face diagonals. --Note: All atoms are identical; the face-centered atoms are shaded

differently only for ease of viewing.

FACE CENTERED CUBIC STRUCTURE (FCC)

ex: Al, Cu, Au, Pb, Ni, Pt, Ag

4 atoms/unit cell: (6 face x ½) + (8 corners x 1/8)

• APF for a face-centered cubic structure = 0.74

ATOMIC PACKING FACTOR: FCC

The maximum achievable APF!

APF =

4

3 p ( 2 a/4 ) 3 4

atoms

unit cell atom

volume

a 3

unit cell

volume

Close-packed directions:

length = 4R = 2 a

Unit cell contains: 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cell a

2 a (a = 22*R)

SECOND CRYSTAL CLASS

TETRAGONAL CRYSTAL CLASS

Symmetry– One 4 fold

4…Simple tetragonal

5…Body centered tetragonal

a = b not equal to c

α = β = γ = 90°

Two parameters to define a and c.

One might suppose stretching face-centered cubic would result in face-centered tetragonal, but face-centered tetragonal is equivalent to body-centered tetragonal, BCT (with a smaller lattice spacing). BCT is considered more fundamental, so that is the standard terminology

THIRD CRYSTAL CLASS

ORTHORHOMBIC CRYSTAL CLASS

Symmetry–Three 2 fold

6…Simple

7…Body centered

8…Face centered

9…End centered

a≠ b ≠ c

α = β = γ = 90°

Three parameters to define a

and c.

4. HEXAGONAL CLOSE-PACKED STRUCTURE

(HCP)

Symmetry–One 6 fold

10…Simple Hexagonal

a = b ≠ c

α = β = 90°

γ = 120°

Two parameters to define a and c.

• Coordination # = 12

• ABAB... Stacking Sequence

• APF = 0.74

• 3D Projection • 2D Projection

4. HEXAGONAL CLOSE-PACKED STRUCTURE

(HCP)4

6 atoms/unit cell

ex: Cd, Mg, Ti, Zn

c

a

A sites

B sites

A sites Bottom layer

Middle layer

Top layer

FIFTH CRYSTAL CLASS

Rhombohedral crystal class

11…Simple rhombohedral

Symmetry– One 3 fold (120°)

a = b = c

α = β = γ ≠ 90°

Two parameters to define a and angle α.

SIXTH CRYSTAL CLASS

Monoclinic crystal class

Symmetry– One 2 fold

12… Simple monoclinic

13…End centered Monoclinic (A and B not C)

a ≠ b ≠ c

α = β = 90° ≠ γ

Four parameters to define a ,b ,c and angle γ.

SEVENTH CRYSTAL CLASS

Triclinic crystal class

14… Simple triclinic

a ≠ b ≠ c

α ≠ β ≠ γ ≠ 90°

Six parameters to define a ,b ,c and anglesα,β ,γ.

HIGHLY UNSYMMETRIC CRYSTAL

THEORETICAL DENSITY, R

where n = number of atoms/unit cell

A = atomic weight

VC = Volume of unit cell = a3 for cubic

NA = Avogadro’s number

= 6.023 x 1023 atoms/mol

Density = =

VC NA

n A =

Cell Unit of Volume Total

Cell Unit in Atoms of Mass

CRYSTALLOGRAPHIC DIRECTIONS, AND

PLANES

Now that we know how atoms arrange themselves to

form crystals, we need a way to identify directions and

planes of atoms

Why? Deformation under loading (slip) occurs on certain

crystalline planes and in certain crystallographic

directions. Before we can predict how materials fail,

we need to know what modes of failure are more likely

to occur.

Other properties of materials (electrical conductivity,

thermal conductivity, elastic modulus) can vary in a

crystal with orientation

MILLER INDICES

USED TO SPECIFY DIRECTIONS AND PLANES.

Vectors and atomic planes in a crystal lattice can be described by a three-value Miller index notation (hkl). The h, k, and l directional indices are separated by 90°, and are thus orthogonal.

Choice of origin arbitrary

Axes and their sense--once chosen not changed and choice is arbitrary.

Unit distance need not to be unity, it is distance between two lattice points in a particular direction in a particular crystal.

CRYSTAL DIRECTIONS

A crystallographic direction is defined as a line between two

points, or a vector.

CRYSTAL DIRECTIONS

Smallest integer reduction.

No separator used

Separator can be used when directions came with

more than one digit number, example [1,14,7]

Bar is used for negative directions.

Characteristics of crystal in different directions is

different that is why it is needed to give them

different name.

Negative Directions

HCP CRYSTALLOGRAPHIC

DIRECTIONS

1. Vector repositioned (if necessary) to

pass

through origin.

2. Read off projections in terms of unit

cell dimensions a1, a2, a3, or c

3. Adjust to smallest integer values

4. Enclose in square brackets, no commas

[uvtw]

[ 1120 ] ex: ½, ½, -1, 0 =>

dashed red lines indicate

projections onto a1 and a2 axes a1

a2

a3

-a3

2

a 2

2

a 1

- a3

a1

a2

z

FAMILY OF DIRECTIONS Directions which look physically

identical not parallel, if it is parallel it

is same direction

o<011>

Permuting it will give 12 different

arrangements

[110]

[1-10]

[-110]

[011]

[101]

[0-11]

[01-1]

[-101]

[10-1]

[-1-10]

[0-1-1]

[-10-1]

ex: linear density of Al in [110]

direction

a = 0.405 nm

LINEAR DENSITY – CONSIDERS EQUIVALENCE AND IS

IMPORTANT IN SLIP

Linear Density of Atoms LD =

a

[110]

Unit length of direction vector

Number of atoms

# atoms

length

1 3.5 nm a 2

2 LD

- = =

DEFINING 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.

Algorithm (in cubic lattices this is direct) 1. 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 i.e., (hkl) families {hkl}

CRYSTAL PLANES

CRYSTALLOGRAPHIC PLANES (HCP)

In hexagonal unit cells the same idea is

used

example a1 a2 a3 c

4. Miller-Bravais Indices (1011)

1. Intercepts 1 -1 1 2. Reciprocals 1 1/

1 0

-1

-1

1

1

3. Reduction 1 0 -1 1

a2

a3

a1

z

FAMILY OF PLANE

All planes which are physical identical is that

arrangement of atoms is same

They are also not parallel if parallel this is same

plane.

PLANAR DENSITY OF (100) IRON Solution: At T < 912C iron has the BCC structure.

(100)

Radius of iron R = 0.1241 nm

R 3

3 4 a =

2D repeat unit

= Planar Density = a 2

1

atoms

2D repeat unit

= nm2

atoms 12.1

m2

atoms = 1.2 x 1019

1

2

R 3

3 4 area

2D repeat unit

COMPARISON OF FCC, HCP, AND

BCC CRYSTAL STRUCTURES

Both FCC and HCP structures are close packed

APF = 0.74.

The closed packed planes are the {111} family for

FCC and the (0001) plane for HCP.

Stacking sequence is ABCABCABC in FCC and

ABABAB in HCP.

BCC is not close packed, APF = 0.68. Most

densely packed planes are the {110} family.

ATOMIC DENSITIES

WHY DO WE CARE?

Properties, in general, depend on linear and planar

density.

Examples:

Speed of sound along directions

Slip (deformation in metals) depends on linear &

planar density. Slip occurs on planes that have the

greatest density of atoms in direction with highest

density (we would say along closest packed directions

on the closest packed planes)

Symbol Alternate

symbols

Direction [ ] [uvw] → Particular direction

< > <uvw> [[ ]] → Family of directions

Plane ( ) (hkl) → Particular plane

{ } {hkl} (( )) → Family of planes

Point . . .abc. [[ ]] → Particular point

: : :abc: → Family of point

POLYMORPHISM: ALSO IN METALS Two or more distinct crystal structures for the

same material (allotropy/polymorphism)

titanium

(HCP), (BCC)-Ti

BCC

FCC

BCC

1538ºC

1394ºC

912ºC

-Fe

-Fe

-Fe

liquid

iron system:

•Carbon (diamond, graphite,)

•Silica (quartz, tridymite,

cristobalite, etc.)

•Iron (ferrite, austenite)

Anisotropic – Direction dependent properties

Example carbon fibers, whiskers

Isotropic – Direction independent properties

Each grain may be anisotropic ,but a specimen

composed by grain aggregate behaves isotropically.

The degree of anisotropy increases with decreasing

structural symmetry. So TRICLINICS are highly

anisotropic

LOOKING AT THE CERAMIC UNIT CELLS

CERAMIC CRYSTAL STRUCTURE

Broader range of chemical composition than metals with more complicated structures

Contains at least 2 and often 3 or more atoms.

Usually compounds between metallic ions (e.g. Fe,Ni, Al) called cations and non-metallic ions (e.g.O, N, Cl) called anions.

How do Cations and Anions arrange themselves in space???

Structure is determined by two characteristics:

1. Electrical charge

Crystal (unit cell) must remain electrically neutral .Sum of cation and anion charges in cell is 0

2. Relative size of the ions

CERAMIC CRYSTAL STRUCTURE

AX-TYPE CRYSTAL STRUCTURES

Some of the common ceramic materials are those in which there are equal numbers

of cations and anions.

Example– NaCl, MgO,Feo,CsCl,Zinc blende

Am Xp -TYPE CRYSTAL STRUCTURES

Not equal number of charges

Example – Fluorite CaF2, UO2, ThO2

Am Bn Xp -TYPE CRYSTAL STRUCTURES

Barium titanate BaTiO3

CESIUM CHLORIDE (CSCL) UNIT CELL SHOWING

(A) ION POSITIONS AND THE TWO IONS PER LATTICE POINT AND

SODIUM CHLORIDE (NACL) STRUCTURE SHOWING

(A) ION POSITIONS IN A UNIT CELL,

(B) FULL-SIZE IONS, AND

(C) MANY ADJACENT UNIT CELLS.

FLUORITE (CAF2) UNIT CELL SHOWING

(A) ION POSITIONS AND

(B) FULL-SIZE IONS.

ARRANGEMENT OF POLYMERIC CHAINS IN THE UNIT CELL

OF POLYETHYLENE. THE DARK SPHERES ARE CARBON ATOMS,

AND THE LIGHT SPHERES ARE HYDROGEN ATOMS. THE UNIT-

CELL DIMENSIONS ARE 0.255 NM × 0.494 NM × 0.741 NM.

DENSITIES OF MATERIAL CLASSES

metals > ceramics > polymers

Why?

(g

/cm

)

3

Graphite/ Ceramics/ Semicond

Metals/ Alloys

Composites/ fibers

Polymers

1

2

2 0

30

*GFRE, CFRE, & AFRE are Glass, Carbon, & Aramid Fiber-Reinforced Epoxy composites (values based on 60% volume fraction of aligned fibers

in an epoxy matrix). 10

3

4

5

0.3

0.4

0.5

Magnesium

Aluminum

Steels

Titanium

Cu,Ni

Tin, Zinc

Silver, Mo

Tantalum Gold, W Platinum

G raphite

Silicon

Glass - soda Concrete

Si nitride Diamond Al oxide

Zirconia

H DPE, PS PP, LDPE

PC

PTFE

PET PVC Silicone

Wood

AFRE *

CFRE *

GFRE*

Glass fibers

Carbon fibers

A ramid fibers

Metals have... • close-packing

(metallic bonding)

• often large atomic masses

Ceramics have... • less dense packing

• often lighter elements

Polymers have... • low packing density

(often amorphous)

• lighter elements (C,H,O)

Composites have... • intermediate values

In general

THANKYOU

ANY QUESTION