Crystalography

Complied by:

Prof. Vijaya Agarwala BE, MTech, PhD

Professor and Head, Center of Excellence Nanotechnology

Professor, Metallurgical and Materials Engineering and

IIT Roorkee

MT201B: Materials Science

L-3, T-1, P-04 credits: CWS-25%, MTE-25%, ETE-50%

Materials Science 2

lntroduction to Crystallography:

Crystal defects: point defects, line defects, dislocations surface defects and volume defects;

Principles of Alloy Formation : primary and intermediate phases, their formation, solid solutions, Hume Rothery rules,

Binary Equilibria: Binary phase diagrams involving isomorphous, eutectic, peritectic and eutectoid reactions. phase rule, lever rule, effect of non-equilibrium cooling on structure and distribution of phases. Some common binary phase diagrams viz : Cu-Ni, Al-Si, Pb-Sn, Cu-Zn, Cu-Sn and Fe-C and important alloys belonging to these systems;

The shell model of the atom in which electrons are confined to live within certain shells and in subshells within shells

Fig 1.3

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Force is considered the change in potential energy, E, over a change in position.

F = dE/dr

Fig 1.8

The formation of ionic bond between Na and Cl atoms in NaCl. The attractionIs due to coulombic forces.

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Fig 1.10

Sketch of the potential energy per ion-pair in solid NaCl. Zero energy corresponds to neutral Na and Cl atoms infinitely separated.

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Materials Science Fig 1.12

The origin of van der Walls bonding between water molecules.(a) The H2O molecule is polar and has a net permanent dipole moment (b) Attractions between the various dipole moments in water gives rise to van der Walls bonding

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Covalent bonding-sharing of electron-strong bond, so high MP-directional, low electrical conductivity

Metallic Bonding-random movements of electron, electron cloud-high electrical conductivity

Crystal Systems

• Most solids are crystalline with their atoms arranged in aregular manner.• Long-range order : the regularity can extend throughout thecrystal.• Short-range order : the regularity does not persist overappreciable distances. eg. amorphous materials such as glassand wax.• Liquids have short-range order, but lack long-range order.• Gases lack both long-range and short-range order

Ref: http://me.kaist.ac.kr/upload/course/MAE800C/chapter2-1.pdf

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Crystal Structures (Contd…)

• Five regular arrangements of lattice points that can occur in two dimensions.

(a) square; (b) primitive rectangular;

(c) centered rectangular; (d) hexagonal;

(e) oblique.

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Point lattice

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Unit cell

Lattice parameters: a, b, c, α, β and γ

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Crystal systems and Crystal systems and Bravais latticeBravais lattice

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Number of lattice points per cell

Where,Ni = number of interior points,Nf = number of points on faces,Nc = number of points on corners.

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base-centered arrangementof points is not a new lattice

Tetragonal unit cell

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Any of the fourteen Bravais lattices may be referred to a

combinatin of primitive unit cells.Face centered cubic latticeshown may be referred to the primitive cubic cell and rhombohedral cell (indicated by dashed lines, its axial angle between a is 600, and each of its side is √2 a, where a is the lattice parameter of cubic cell.

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000, ½ ½ 0, ½ 0 ½ , 0 ½ ½ ¼ ¼ ¼ , ¾ ¾ ¼, ¾ ¼ ¾, ¼ ¾ ¾

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A C G H

D F I J

000, ½ ½ 0, ½ 0 ½ , 0 ½ ½ ¼ ¼ ¼ , ¾ ¾ ¼, ¾ ¼ ¾, ¼ ¾ ¾

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3a 3b 5a 5b

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Graphite

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Fig 1.43Three allotropes of carbon

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Coordination numberNumber of nearest neighbors of an atom in the crystal lattice

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• Rare due to poor packing (only Po has this structure)• Close-packed directions are cube edges.

• Coordination # = 6 (# nearest neighbors)

(Courtesy P.M. Anderson)

SIMPLE CUBIC STRUCTURE (SC)SIMPLE CUBIC STRUCTURE (SC)

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Polonium is a chemical element with the symbol Po and atomic number 84, discovered in 1898 by Marie and Pierre Curie. A rare and highly radioactive element ...

APF = Volume of atoms in unit cell*

Volume of unit cell

*assume hard spheres

• APF for a simple cubic structure = 0.52

APF = a3

3(0.5a)31

unit cellatom

volume

unit cellvolume

close-packed directions

R=0.5a

contains 8 x 1/8 = 1 atom/unit cell

Adapted from Fig. 3.19, Callister 6e.

ATOMIC PACKING FACTORATOMIC PACKING FACTOR

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• Coordination # = 8

Adapted from Fig. 3.2, Callister 6e.

• Close packed directions are cube diagonals.

--Note: All atoms are identical; the center atom is shaded differently only for ease of viewing.

BODY CENTERED CUBIC STRUCTURE (BCC)BODY CENTERED CUBIC STRUCTURE (BCC)

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• APF for a body-centered cubic structure = 0.68

Close-packed directions: length = 4R

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

Adapted fromFig. 3.2,Callister 6e.

ATOMIC PACKING FACTOR: BCCATOMIC PACKING FACTOR: BCC

APF = a3

3( 3a/4)32

unit cell atomvolume

unit cell

volume

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• Coordination # = 12

Adapted from Fig. 3.1(a), Callister 6e.

• Close packed directions are face diagonals.

--Note: All atoms are identical; the face-centered atoms are shaded differently only for ease of viewing.

FACE CENTERED CUBIC STRUCTURE (FCC)FACE CENTERED CUBIC STRUCTURE (FCC)

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APF = a3

3( 2a/4)34

unit cell atomvolume

unit cell

volume

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

• APF for a body-centered cubic structure = 0.74

Close-packed directions: length = 4R

Adapted fromFig. 3.1(a),Callister 6e.

ATOMIC PACKING FACTOR: FCCATOMIC PACKING FACTOR: FCC

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Example: Copper

n AVcNA

# atoms/unit cell Atomic weight (g/mol)

Volume/unit cell

(cm3/unit cell)Avogadro's number (6.023 x 1023 atoms/mol)

Data from Table inside front cover of Callister (see next slide):• crystal structure = FCC: 4 atoms/unit cell• atomic weight = 63.55 g/mol (1 amu = 1 g/mol)• atomic radius R = 0.128 nm (1 nm = 10 cm)-7

Vc = a3 ; For FCC, a = 4R/ 2 ; Vc = 4.75 x 10-23cm3

Compare to actual: Cu = 8.94 g/cm3Result: theoretical Cu = 8.89 g/cm3

THEORETICAL DENSITY, THEORETICAL DENSITY,

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Element Aluminum Argon Barium Beryllium Boron Bromine Cadmium Calcium Carbon Cesium Chlorine Chromium Cobalt Copper Flourine Gallium Germanium Gold Helium Hydrogen

Symbol Al Ar Ba Be B Br Cd Ca C Cs Cl Cr Co Cu F Ga Ge Au He H

At. Weight (amu) 26.98 39.95 137.33 9.012 10.81 79.90 112.41 40.08 12.011 132.91 35.45 52.00 58.93 63.55 19.00 69.72 72.59 196.97 4.003 1.008

Atomic radius (nm) 0.143 ------ 0.217 0.114 ------ ------ 0.149 0.197 0.071 0.265 ------ 0.125 0.125 0.128 ------ 0.122 0.122 0.144 ------ ------

Density (g/cm3) 2.71 ------ 3.5 1.85 2.34 ------ 8.65 1.55 2.25 1.87 ------ 7.19 8.9 8.94 ------ 5.90 5.32 19.32 ------ ------

Crystal Structure FCC ------ BCC HCP Rhomb ------ HCP FCC Hex BCC ------ BCC HCP FCC ------ Ortho. Dia. cubic FCC ------ ------

Adapted fromTable, "Charac-teristics ofSelectedElements",inside frontcover,Callister 6e.

Characteristics of Selected Elements at 20C

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metals ceramics polymers

Graphite/ Ceramics/ Semicond

Metals/ Alloys

Composites/ fibersPolymers

30Based on data in Table B1, Callister *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

0.3 0.4 0.5

Magnesium

Aluminum

Steels

Titanium

Tin, Zinc

Silver, Mo

Tantalum Gold, W Platinum

Graphite Silicon

Glass -soda Concrete

Si nitride Diamond Al oxide

Zirconia

HDPE, PS PP, LDPE

PET PVC Silicone

AFRE *

CFRE *

Glass fibers

Carbon fibers

Aramid fibers

Metals have... • close-packing (metallic bonding) • large atomic mass Ceramics have... • less dense packing (covalent bonding) • often lighter elements Polymers have... • poor packing (often amorphous) • lighter elements (C,H,O) Composites have... • intermediate values

Data from Table B1, Callister 6e.

DENSITIES OF MATERIAL CLASSESDENSITIES OF MATERIAL CLASSES

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Physical Properties•Acoustical properties•Atomic properties•Chemical properties•Electrical properties•Environmental properties•Magnetic properties•Optical properties•Density

Mechanical properties•Compressive strength•Ductility•Fatigue limit•Flexural modulus•Flexural strength•Fracture toughness•Hardness•Poisson's ratio•Shear modulus•Shear strain•Shear strength•Softness•Specific modulus•Specific weight•Tensile strength•Yield strength•Young's modulus

• Most engineering materials are polycrystals.

• Nb-Hf-W plate with an electron beam weld.• Each "grain" is a single crystal.• If crystals are randomly oriented, overall component properties are not directional.• Crystal sizes typ. range from 1 nm to 2 cm (i.e., from a few to millions of atomic layers).

Adapted from Fig. K, color inset pages of Callister 6e.(Fig. K is courtesy of Paul E. Danielson, Teledyne Wah Chang Albany)

POLYCRYSTALSPOLYCRYSTALS

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• Single Crystals

-Properties vary with direction: anisotropic.

-Example: the modulus of elasticity (E) in BCC iron:

• Polycrystals

-Properties may/may not vary with direction.-If grains are randomly oriented: isotropic. (Epoly iron = 210 GPa)-If grains are textured, anisotropic.

E (diagonal) = 273 GPa

E (edge) = 125 GPa

Data from Table 3.3, Callister 6e.(Source of data is R.W. Hertzberg, Deformation and Fracture Mechanics of Engineering Materials, 3rd ed., John Wiley and Sons, 1989.)

Adapted from Fig. 4.12(b), Callister 6e.(Fig. 4.12(b) is courtesy of L.C. Smith and C. Brady, the National Bureau of Standards, Washington, DC [now the National Institute of Standards and Technology, Gaithersburg, MD].)

SINGLE VS POLYCRYSTALSSINGLE VS POLYCRYSTALS

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Face-Centered Cubic Nanoparticles

• Figure (a) shows the 12 neighbors that surround an atom(darkened circle) located in the center of a cube for a FCC lattice.• Figure (b) presents another perspective of the 12 nearest neighbors.These 13 atoms constitute the smallest theoretical nanoparticle for anFCC lattice.• Figure (c) shows the 14-sided polyhedron, called adekatessarahedron, that is generated by connecting the atoms with

planer faces

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If another layer of 42 atoms is layed around the 13-atom nanoparticle, one obtains a 55-atom nanoparticle with the same dekatessarahedron shape.

Lager nanoparticles with the same polyhedral shape are obtained by adding more layers, and thesequence of numbers in the resulting particles, N

N=1, 13, 55, 147,.., which are called structural magic numbers.

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Atoms in nano clusters

• For n layers, the number of atoms N and the number of atoms on the surface Nsurf in this FCC nanoparticle is given by the formula,

N = 1/3(10 n3 −15 n2 +11 n −3)

Nsurf =10n2 − 20n +12

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Atomic packing• In two dimensions the most efficient way to pack identical circles is equilateral triangle

arrangement shown in figure (a).

• A second hexagonal layer of spheres can be placed on top of the first to form the most efficient packing of two layers, as shown in figure (b).

• For efficient packing, the third layer can be placed either above the first layer with an atom at the location indicated by T or in the third possible arrangement with an atom above the position marked by X on the figure.

• In the first case a hexagonal lattice with a hexagonal close packed (HCP) structure is generated, and in the second case a face-centered cubic lattice results.

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X on figure is called an octahedral site

The radius(aoct) of octahedral site is = 0.41421ao

where ao is the radius of the spheres.

There are also smaller sites, called tetrahedral sites, labeled T

This is a smaller site since its radius aT= 0.2247ao

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Void types

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Stacking sequences: FCC & HCP

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FCC stacking sequence

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Plan-stacking sequence

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HCP structure

Stacking sequence

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Fig 1.40

Miller In

dex- dire

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Lattice directions- MI

The direction of any line in a lattice

may be described by first drawing a line through the origin parallel

to the given line and then giving the coordinates of any point on the line

through the origin.

-smallest integer value

- Negative directions are shown by bars eg.

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Plane designation by Miller indices

-Miller indices are always cleared of fractions

- If a plane is parallel to a given axis, its fractional intercept on that

axis is taken as infinity, Miller index is zero

- If a plane cuts a negative axis, the corresponding index is negative and is written with a bar over it.

-Planes whose indices are the negatives of one another are parallel and lie on opposite sides of the origin, e.g., (210) and (-2ī0).

-- Planes belonging to the same family is denoted by curly bracket , {hkl}

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Fig 1.41

Labeling of crystal planes and typical examples in the cubic lattice

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Miller indices of lattice planes

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Miller Index

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The hexagonal unit cell : Miller –Bravais indices of planes and directions

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Zone= zonal planes + zonal axis

-Zone axis and (hkl) the zonal plane

All shaded planes belong to the same zone i.e parallel to an axis called zone axsis64Materials Science

u v wh1 k1 l1h2 k2 l2

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Crystal defects

1.Point defect- Vacancy, Impurity atoms ( substitutional and interstitial) Frankel and Schottky defect ( ionic solids & nonstochiometric)

2. Line defect- Edge dislocation Screw dislocation, Mixed dislocation

3. Surface defects- Grain boundaries Twin boundary Surfaces, stacking faults Interphases

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Vacancy

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Impurity

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Frankel and Schottky defect

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Non stochiometry

Conduction in ionic crystal

ZnO crystal containing extra Zn2+

Crystal is electronically neutral, (i.e. 2+ & 2- )

electron

The mobile 2 electrons

give rise to the

ionic conduction

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Dislocation line and b are perpendicular to each other

Line defects

1.Edge dislocation

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Movement of edge dislocation

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Cause of slip

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Elastic stress field responsible for electron scattering and increase in electrical resistivity

lattice strain around dislocation

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When dislocations move?

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The closest packed plane and the closest packed direction of FCC

The plane and directions for the dislocation movement

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Tensile specimen- breaks

How does the dislocationaffect the failure?

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Line defects

2. Screw dislocation

Dislocation line and b are parallel to each other 84Materials Science

Line defects

3. Mixed dislocatio

By resolving, the contribution from both types of dislocations can be determined

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TEM-dislocaions

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Grain boundaries

3. Surface defects

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Low angle GB

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Twin boundary

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Stacking fault-occurs when there is a flaw in the stacking sequence

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Interfaces of phases

Coherent semi-coherent incoherent

Al-Cu system

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Principles of Alloy Formation :

primary and intermediate phases,their formation, solid solutions,Hume Rothery rule

Definition of Phase:• A phase is a region of material that is chemically

uniform, physically distinct, and (often) mechanically separable.

• A phase is a physically separable part of the system with distinct physical and chemical properties. System - A system is that part of the universe which is under consideration.

• In a system consisting of ice and water in a glass jar, the ice cubes are one phase, the water is a second phase, and the humid air over the water is a third phase. The glass of the jar is another separate phase.

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Gibbs' phase rule proposed by Josiah Willard Gibbs The phase rule is an expression of the number of variablesin equation(s) that can be used to describe a system in equilibrium.

Degrees of freedom, F

F = C − P + 2 F = C − P + 2

Where, P is the number of phases in thermodynamic equilibrium with each other C is the number of components

Phase rule at constant pressure Phase rule at constant pressure

• Condensed systems have no gas phase. When their properties are insensitive to the (small) changes in pressure, which results in the phase rule at constant pressure as,

F = C − P + 1

Types of Phase diagram

1. Unary phase diagram2. Binary phase diagrams3. Ternary phase diagram

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Unary phase diagram

Critical pressure Liquid phase

Temperature

Solid Phase gaseous phase

Binary phase diagrams

1. Binary isomorphous systems (complete solid solubility)

2. Binary eutectic systems (limited solid solubility)

3. Binary systems with intermediate phases/compounds

Binary phase diagram- isomorphous system

The Lever RuleFinding the amounts of phases in a two phase region:1. Locate composition and temperature in diagram2. In two phase region draw the tie line or isotherm3. Fraction of a phase is determined by taking thelength of the tie line to the phase boundary for theother phase, and dividing by the total length of tielineThe lever rule is a mechanicalanalogy to the mass balancecalculation. The tie line in thetwo-phase region is analogous toa lever balanced on a fulcrum.

microstrucures

Binary phase diagram –2. limited solubility

• A phase diagram for a binary system displaying an eutectic point.

Cu-Ag system

Sn-Bi system

Pb-Sn system

Mechanismof growth

Pb-Sn system

Fig 1.69

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The equilibrium phase diagram of the Pb-Sn alloy.

The microstructure on the left show the observations at various points during the coolingof a 90% Pb-10% Sn from the melt along the dashed line (the overall alloy composition remains constant at 10% Sn).

Pb-Sn system

Cu- Zn system

Ternary phase diagrams

MgO-Al2O3-SiO2 system at 1 atm. pressure Fe-Ni-Cr ternary alloy system

Formation of nano crystallites/ grains

Nuclei of the solid phase form and they grow to consume all the liquid at the solidus line. 13 atoms constitute to a theoretical nanoparticle for a FCC lattice having two layers. 55 and 147 atoms for 3 and 4 layer clusters. If the size of the crystallites are in the nanometer range, they are called nanocrystals/grains.

High temperature structure can be retained at lowertemperature by quenching.

Single crystal

A single crystal solid is a material in which the crystal lattice of the entire sample is continuous

no grain boundaries- grain boundaries can have significant effects on the physical and electrical properties of a material

single crystals are of interest to electric device applications

Crystalography

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nacl materials science

materials science polonium

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amorphous materials

crystal lattice

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