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Ch 3:CRYSTAL STRUCTURES &

PROPERTIES(adapted from Callister)

Mir M. Atiqullah, Ph.D.

Focus for this chapter:

• How do atoms assemble into solid structures?(e.g. metals)- even molecules of compounds

• How does the density of a material depend onits structure?

• When do material properties vary with thetest sample (i.e., part) orientation/direction?

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Ch3: Crystal Structures

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• Non dense, random packing

• Dense, regular packing

Dense, regular-packed structures tend to havelower energy. = stronger bonding!

Energy

r

typical neighbor bond length

typical neighbor bond energy

Energy

r

typical neighbor bond length

typical neighbor bond energy

ENERGY AND PACKING- (motivation for)

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• atoms pack in periodic, 3D arrays• typical of:

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Crystalline materials...

-metals-many ceramics-some polymers (partial)

• atoms have no periodic packing• occurs for:

Noncrystalline materials...

-complex structures-by rapid solidification

Si Oxygen

crystalline SiO2

noncrystalline SiO2"Amorphous" = NoncrystallineAdapted from Fig. 3.18(b),Callister 6e.

Adapted from Fig. 3.18(a),Callister 6e.

What is a crystalline Material ?

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• tend to be densely packed.

• have several reasons for dense packing:-Typically, only one element is present, so all atomic radii are the same.-Metallic bonding is non-directional.-Nearest neighbor distances tend to be small in order to lower bond energy.

• have the simplest crystal structures.

We will look closely at four (4) such structures...

METALLIC CRYSTALS

A Unit Crystal CellCrystal Lattice & Lattice Constants

• Corners represent atomic sites/positions.

• a,b,c,α,β,γ are lattice constant that define the lattice structure

Unit cell: Smallest packing pattern of the atoms that repeats…

Crystal lattice- a 3 D arrangement of points representing atomic positions

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Unit cell geometries

Unit cell= smallest repetitive

Pattern/stacking

These are

7 Basic types

14 variations possible

Here are the 14 varieties

What is the difference between crystal structure and crystal system?

Answer: A crystal structure is described by both the geometry of, and atomic arrangements within, the unit cell, whereas a crystal system is described only in terms of the unit cell geometry. For example, face-centered cubic and body-centered cubic are crystal structures that belong to the cubiccrystal system.

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5

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

• Coordination # = 6(# of nearest neighbors)

(Courtesy P.M. Anderson)

SIMPLE CUBIC STRUCTURE (SC)

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APF = Volume of atoms in unit cell*

Volume of unit cell

*assume hard spheres

• APF for a simple cubic structure = 0.52

APF = a3

4

3π (0.5a)31

atoms

unit cellatom

volume

unit cellvolume

close-packed directions

a

R=0.5a

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

Adapted from Fig. 3.19,Callister 6e.

ATOMIC PACKING FACTOR-Simple Cubic

• Coordination # = 8

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Adapted from Fig. 3.2,Callister 6e.

(Courtesy P.M. Anderson)

• 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 Crystal (BCC)

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aR

8

• APF for a body-centered cubic structure = 0.68

Close-packed directions: length = 4R

= 3 a

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

Adapted fromFig. 3.2,Callister 6e.

ATOMIC PACKING FACTOR: BCC

APF = a3

4

3π ( 3a/4)32

atoms

unit cell atomvolume

unit cell

volume

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• Coordination # = 12

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

(Courtesy P.M. Anderson)

• 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)

APF = a3

4

3π ( 2a/4)34

atoms

unit cell atomvolume

unit cell

volume

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

a

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• For a face-centered cubic structure APF = 0.74

Close-packed directions: length = 4R

= 2 a

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

ATOMIC PACKING FACTOR: FCC

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• ABCABC... Stacking Sequence• 2D Projection

A sites

B sites

C sitesB B

B

BB

B BC C

CA

A

• FCC Unit CellA

BC

FCC STACKING SEQUENCE

12

• Coordination # = 12

• ABAB... Stacking Sequence

• APF = 0.74

• 3D Projection

A sites

B sites

A sites

Bottom layer

Middle layer

Top layer

Adapted from Fig. 3.3,Callister 6e.

HEXAGONAL CLOSE-PACKED STRUCTURE (HCP)

Atoms per unit HCP cell

= 3 + 12* 1/6 + 2* ½ = 6

APF = 0.74 ( same as FCC)

Challenge- Show this.

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• Compounds: Often have similar close-packed structures.

• Close-packed directions--along cube edges.

• Structure of NaClWhich color is Na ?

(Courtesy P.M. Anderson) (Courtesy P.M. Anderson)

STRUCTURE OF COMPOUNDS: NaCl

Size of compound mol affects the cell size

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Some Crystal molecules

diamond

Graphite – Hexagonal

SiO4-group Structural unit of Quartz

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ρ = n AVcNA

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

Volume/unit cell

(cm3/unit cell)Avogadro's number

(6.023 x 1023 atoms/mol)

THEORETICAL DENSITY, ρ

Balancing units:

... volg

atomsmole

volcell

moleg

cellatoms

moleatoms

cellvol

moleg

cellatoms

=⋅⋅⋅=⋅

⋅=ρ

Given Crystal structure, atomic wt.,density!VC!R( using ?)

Example: Density of Copper

• 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)

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

Result: theoretical ρCu = 8.89 g/cm3

Data from Table inside front cover of Callister (see next slide):

Compare to actual: ρCu = 8.94 g/cm3

Why the difference? Existence of Isotopes- some atoms with more neutrons in nucleus.

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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

ρmetals ρceramics ρpolymers

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ρ (g

/cm

3)

Graphite/ Ceramics/ Semicond

Metals/ Alloys

Composites/ fibersPolymers

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2

20

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

3 4 5

0.3 0.4 0.5

Magnesium

Aluminum

Steels

Titanium

Cu,Ni

Tin, Zinc

Silver, Mo

Tantalum Gold, W Platinum

Graphite Silicon

Glass-soda Concrete

Si nitride Diamond Al oxide

Zirconia

HDPE, PS PP, LDPE

PC

PTFE

PET PVC Silicone

Wood

AFRE*

CFRE*

GFRE*

Glass fibers

Carbon fibers

Aramid fibers

Why?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 CLASSES

Polymorphism or Allotropy

Some metals/nonmetals may exist in more than one crystal structure depending on both temperature and pressure.Expl: Carbon ! graphite, diamond, Bucky Ball C60 nanotube O2 O3

BCC iron ! high temp. ! FCC ironThis condition of multiple crystal structures is also known as ALLOTROPY.See p-48 on Tin-Disease" Tin (Sn) at room Temp BCC. At T<13.2 C, it transforms into

a structure similar to diamond ( complex grid structure) with lower density and larger volume. May cause disintegration/crumbling ... Read the story of Russian disaster in 1850.

See the full story in Text

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Review :

Polymophism --Difference between theoretical and actual density of materials—Interstitial –Crystal structure vs Crystal systemHow many crystal system, structureAPFCoordination numberClose packed direction, planeSlip plane

Crystallographic Directions: Cubic cellA cubic cell with usual coordinates

Example directions = vectors with lattice constants (side of the cube) =1

Direction of Red arrow:

Temporary new origin O’ (if needed, translate the vector in parallel so that it starts at origin or at a corner and contained within unit cell)

Direction vector –1, 1, 0.

Clear fraction : -1, 1, 0.

Format :

O’

0] 1 1[Example direction: Green Arrow.Temporary origin:Direction vector:Clear Fraction:

Format:

O”½, 1, -11, 2, -2

O”

]2 2 1[

½

HCP Directions

a1, a2, a3, z ! 4 directions/coordinates

Same Method/procedure as cubic:

1. Vector in units of coordinates a1, a2, a3, z

2. Clear fraction

3. Format

Example: Red arrow direction.

Vector: 1 -½ -1½ -1 ?

Clear Fraction: 2 -1 -3 -2 (multiplying by 2)

Formatted: [ ?? ]

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Linear Density ( of atomic Packing)

ex: linear density of Al in [110] directionAl - Lattice Constant a = 0.405 nm

Linear Density of Atoms ≡ LD =Unit length of direction vector

Number of atoms

# atoms

length

13.5 nma2

2LD −==

a

[110]

Directions having same linear densities are considered EQUIVALENT DIRECTIONS.

Al crystal- BCC, HCP, FCC, SC ?

How about along [100] direction?

Crystal PlanesUsing 3 Miller Indices:

Procedure-

1.If plane goes thru origin, choose another parallel plane

2.length of the intercepts in terms of a, b, c.

3.Calc. Reciprocal

4.Change the numbers to smallest integer

5. Indices are (hkl)

Ex: Fig (b)-

2. 1 1 ∞3. 1 1 0 4. (1 1 0)

Ex: A plane in a cubic cell

z

x

ya b

c•

••

4. Miller Indices (634)

example1. Intercepts 1/2 1 3/4

a b c

2. Reciprocals 1/½ 1/1 1/¾2 1 4/3

3. Reduction 6 3 4

(001)(010),

Define: Family of Planes {hkl} ! having same planar density

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

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Example: Planes in cubic cell

Example 3.9: Miller Indices?

Fig. (a) Plane thru origin.

1. Move the plane/origin along y-axis.

2.intercepts: inf., -1, ½

3. Reciprocal: 0, -1, 2

4. Smallest integers: 0, -1, 2

5. Indices are (hkl): ) 2 1 0(

Problem 3.41 Plane A.

1. Move the origin to O’, all intercepts within 1 unit..

2. Intercepts: ½, -½ , infinity.

3. Reciprocal: 2, -2, 0

4. Integers: 2, -2, 0

5. Indices are (hkl):

O’

0) 2 2(

Atomic Arrangements

What is the atomic packing in a given crystallographic plane?Packing of atoms often determine the characteristic of that materials in certain application, e.g. catalyst.Iron foil can be used as a catalyst. The atomic packing of the exposed planes is important.

" Consider (100) and (111) crystallographic planes for Fe." What are the planar densities for each of these planes?

Planar Density: Example =(100) Iron

Solution: At T < 912°C iron has the BCC structure.

(100)

Radius of iron R = 0.1241 nm

R3

34a =

Adapted from Fig. 3.2(c), Callister 7e.

2D repeat unit

= Planar Density =a2

1atoms

2D repeat unit

= nm2

atoms12.1m2

atoms= 1.2 x 10191

2

R3

34area2D repeat unit

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Planar Density of (111) Iron

1 atom in plane/ unit surface cell

333 2

2

R3

16R3

42a3ah2area =

===

atoms in plane

atoms above plane

atoms below plane

ah23=

a2

1= =

nm2atoms7.0

m2atoms0.70 x 1019

3 2R3

16Planar Density =

atoms2D repeat unit

area2D repeat unit

Family of Planes

Family – (most) have common characteristics – same height, same hair, eye color etc.Planes in a family, similarly must have same planar densities.In a given crystal unit cell, all planes having same density belong to the same family.E.g. in a BCC cell a family of planes is {100} = (100), (010), (001), (-100), (101), (0-10), (110), (00-1), (111). 3 are wrongFor a different shaped crystal, this may not be true. Rule of Thumb for Cubic Cell only:" Planes in the family have same indices, +ve or –ve, irrespective of

the sequence, are all equivalent. E.g. {123} includes planes (123), (312), (-231) etc.

REVIEWFamily of planes – all planes have same ____?If a crystalline direction goes thru origin (of the cell coordinates)- what should be done?If a crystal plane goes through the origin, then? _____?

APF =? COORD. # = ?

Review – Cryst. Direction! [ 1 2 1], Miller indices (h k l) ! ( 2

1 2),

family of planes (cubic){ 1 1 0} includes (1 0 1), (1 1 0), (1 0 0), (0 0 1), (0 1 1) (0 1 0), (1 0 1), any plane not in the family?

Direction indices=[ ]

1/3

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• Some engineering applications require single crystals:

• Crystal properties reveal featuresof atomic structure.

(Courtesy P.M. Anderson)

--Ex: Certain crystal planes in quartzfracture more easily than others.

(True for large salt crystals.)

--diamond singlecrystals for abrasives

--turbine blades--silicon wafers

Fig. 8.30(c), Callister 6e.(Fig. 8.30(c) courtesyof Pratt and Whitney).

(Courtesy Martin Deakins,GE Superabrasives, Worthington, OH. Used with permission.)

CRYSTALS AS BUILDING BLOCKS

Solidification leading to polycrystalline structure

Neucleation –forming of first solid

Boundary –formation

Grain boundary Under microscope

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• 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)

1 mm

POLYCRYSTALS

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• Single Crystals- bulk is just one crystal

-Properties vary withdirection: anisotropic.

-Example: the modulusof elasticity (E) in BCC iron:

• Polycrystals

-Properties may/may notvary with direction.

-If grains are randomlyoriented: isotropic.(Epoly iron = 210 GPa)

-If grains are textured,anisotropic.

E (diagonal) = 273 GPa

E (edge) = 125 GPa

200 µm

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 POLYCRYSTALS-Their effects

What is –ANISOTROPY ?

Basically It is the dependence of materials properties on the (crystallographic) direction.

-e.g. strength of wood depends on the fiber orientation

ISOTOPIC MATERIALS- whose properties are INDEPENDENT of direction of measurement..

Polycrystalline Material – (generally) ISOTROPIC

Which of these are ISOTROPIC?: wood, steel, fabric, polymer/plastic, fiberglass fishing rod, rubber, ceramics

X Ray Diffraction (XRD)

Did you know x-ray is the main technology to determine/estimate crystal structure?Why? – x-ray wavelengths ~ lattice constants a, b, c etc. How long are wavelengths of Xray compared to visible light? Microwave? FM ? Let’s look at the comparative chart next !

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Electro magnetic Spectrum

Where is the range for cell phones?

Constructive and destructive Interference of light waves

In phase

Out of phase

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• Incoming X-rays diffract from crystal planes.

• Measurement of:Critical angles, θc,for X-rays provideatomic spacing, d.

Adapted from Fig. 3.2W, Callister 6e.

X-RAYS TO CONFIRM CRYSTAL STRUCTURE

reflections must be in phase to detect signal

spacing between planes

d

incoming

X-rays

outgoin

g X-rays

detector

θλ

θextra distance travelled by wave “2”

“1”

“2”

“1”

“2”

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X-ray Diffraction Calculations-Bragg’s law

222 cells, cubic Also

sin2

law) s(Bragg' sin2distance extra the

lkhadnd

dnQTSQn

hklhkl

hkl

++=

⋅=

⋅⋅=+=

θλ

θλλ

Knowing type of cubic, we can select the PEAK and its angle and the reflecting plane.

(Goniometer in a) Xray Diffractometer

T – source of x ray

C - Detector (goniometer, with angle measurement)

S – powder crystalline sample – with possibly many grains reflecting x-ray.

2Ө = angle of diffraction

d=nλ/2sinθc

x-ray intensity (from detector)

θ

θc

Bragg’s law Demo

How it works?

Material is prepared as fine powder – that still keeps many crystals.X-ray will be reflected from various dense reflecting planes with relatively higher intensity.Angles (2θ) of all these intense reflections are measured using the goniometer (of the diffractometer).

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Typical Diffraction Pattern

Material : Pb (SC, BCC or FCC ?) powder

Type h2+k2+l2

Simple Cubic (All but 7) 1, 2, 3, 4, 5, 6, 8

BCC (Even) 2, 4, 6, 8, 10, 12, 14, 16

FCC 3, 4, 8, 11, 12, 16

Crystals of various materials but same structure, will diffract at different angles but in the same sequence of planes.

TRUE/FALSE ?

• Atoms may assemble into crystalline oramorphous structures.

• We can predict the density of a material,provided we know the atomic weight, atomicradius, and crystal geometry (e.g., FCC,BCC, HCP).

• Material properties generally vary with singlecrystal orientation (i.e., they are anisotropic),but properties are generally non-directional(i.e., they are isotropic) in polycrystals withrandomly oriented grains.

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SUMMARY