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Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

CHAPTER

3 Crystal

and Amorphous Structure

in Materials

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

The Space Lattice and Unit Cells

• Atoms, arranged in repetitive 3-Dimensional pattern, in long range order (LRO) give rise to crystal structure.

• Properties of solids depends upon crystal structure and bonding force.

• An imaginary network of lines, with atoms at intersection of lines, representing the arrangement of atoms is called space lattice.

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

Building blocks

LEGO block

Unit cell

http://loyalkng.com/2010/03/21/lego-ironman-from-brickshelfs-legodreams-the-ironman-goes-lego/

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

Unit cell

• Unit cell is that block of atoms which repeats itself to form space lattice.

• Materials arranged in short range order are called amorphous materials

Unit Cell

Space Lattice

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

Crystal Systems and Bravais Lattice

• Only seven different types of unit cells are necessary to create all point lattices.

• According to Bravais (1811-1863) fourteen standard unit cells can describe all possible lattice networks.

• The four basic types of unit cells are  Simple  Body Centered  Face Centered  Base Centered

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

Types of Unit Cells

• Cubic Unit Cell  a = b = c  α = β = γ = 900

• Tetragonal  a =b ≠ c  α = β = γ = 900

Simple Body Centered

Face centered

Simple Body Centered

Figure 3.2

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

Types of Unit Cells (Cont..)

• Orthorhombic  a ≠ b ≠ c  α = β = γ = 900

• Rhombohedral  a =b = c  α = β = γ ≠ 900

Simple Base Centered

Face Centered Body Centered

Simple

Figure 3.2

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

Types of Unit Cells (Cont..)

• Hexagonal  a ≠ b ≠ c  α = β = γ = 900

• Monoclinic  a ≠ b ≠ c  α = β = γ = 900

• Triclinic  a ≠ b ≠ c  α = β = γ = 900

Simple

Base Centered

Figure 3.2

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

Principal Metallic Crystal Structures

• 90% of the metals have either Body Centered Cubic (BCC), Face Centered Cubic (FCC) or Hexagonal Close Packed (HCP) crystal structure.

• HCP is denser version of simple hexagonal crystal structure.

BCC Structure FCC Structure HCP Structure

Figure 3.3

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

Body Centered Cubic (BCC) Crystal Structure

• Represented as one atom at each corner of cube and one at the center of cube.

• Each atom has 8 nearest neighbors. • Therefore, coordination number is 8. • Examples :-

 Chromium (a=0.289 nm)  Iron (a=0.287 nm)  Sodium (a=0.429 nm)

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

BCC Crystal Structure (Cont..)

• Each unit cell has eight 1/8 atom at corners and 1 full atom at the center.

• Therefore each unit cell has

• Atoms contact each other at cube diagonal

(8x1/8 ) + 1 = 2 atoms

4RTherefore, a =

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

Atomic Packing Factor of BCC Structure

Atomic Packing Factor = Volume of atoms in unit cell Volume of unit cell

Vatoms = = 8.373R3

4  





 



 R = 12.32 R3

Therefore APF = 8.723 R 3

12.32 R3 = 0.68

V unit cell = a3 =

 





 



 Π

4 .2

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

Face Centered Cubic (FCC) Crystal Structure

• FCC structure is represented as one atom each at the corner of cube and at the center of each cube face.

• Coordination number for FCC structure is 12 • Atomic Packing Factor is 0.74 • Examples :-

 Aluminum (a = 0.405)  Gold (a = 0.408)

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

FCC Crystal Structure (Cont..)

• Each unit cell has eight 1/8 atom at corners and six ½ atoms at the center of six faces.

• Therefore each unit cell has

• Atoms contact each other across cubic face diagonal

(8 x 1/8)+ (6 x ½) = 4 atoms

4RTherefore, lattice constant a =

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

Hexagonal Close-Packed Structure

• The HCP structure is represented as an atom at each of 12 corners of a hexagonal prism, 2 atoms at top and bottom face and 3 atoms in between top and bottom face.

• Atoms attain higher APF by attaining HCP structure than simple hexagonal structure.

• The coordination number is 12, APF = 0.74.

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

HCP Crystal Structure (Cont..)

• Each atom has six 1/6 atoms at each of top and bottom layer, two half atoms at top and bottom layer and 3 full atoms at the middle layer.

• Therefore each HCP unit cell has

• Examples:-  Zinc (a = 0.2665 nm, c/a = 1.85)  Cobalt (a = 0.2507 nm, c.a = 1.62)

• Ideal c/a ratio is 1.633.

(2 x 6 x 1/6) + (2 x ½) + 3 = 6 atoms

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

YouTube videos links

• http://www.youtube.com/watch?v=Rm- i1c7zr6Q

• http://www.youtube.com/watch?v=F4Du 4zI4GJ0

• http://www.youtube.com/watch?v=vYky UqUa6vU

• http://www.youtube.com/watch?v=NYV SI83KiKU

http://www.youtube.com/watch?v=Rm-i1c7zr6Q

http://www.youtube.com/watch?v=F4Du4zI4GJ0

http://www.youtube.com/watch?v=vYkyUqUa6vU

http://www.youtube.com/watch?v=NYVSI83KiKU

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

Atom Positions in Cubic Unit Cells

• Cartesian coordinate system is use to locate atoms. • In a cubic unit cell

 y axis is the direction to the right.  x axis is the direction coming out of the paper.  z axis is the direction towards top.  Negative directions are to the opposite of positive directions.

• Atom positions are located using unit distances along the axes.

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

Directions in Cubic Unit Cells

• In cubic crystals, Direction Indices are vector components of directions resolved along each axes, resolved to smallest integers.

• Direction indices are position coordinates of unit cell where the direction vector emerges from cell surface, converted to integers.

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

Procedure to Find Direction Indices

(0,0,0)

(1,1/2,1)

z Produce the direction vector till it emerges from surface of cubic cell

Determine the coordinates of point of emergence and origin

Subtract coordinates of point of Emergence by that of origin

(1,1/2,1) - (0,0,0) = (1,1/2,1)

Are all are integers?

Convert them to smallest possible

integer by multiplying by an integer.

2 x (1,1/2,1) = (2,1,2)

Are any of the direction vectors negative?

Represent the indices in a square bracket without comas with a over negative index (Eg: [121])

Represent the indices in a square bracket without comas (Eg: [212] )

The direction indices are [212]

YES

YES NO

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

Direction Indices - Example

• Determine direction indices of the given vector. Origin coordinates are (3/4 , 0 , 1/4). Emergence coordinates are (1/4, 1/2, 1/2).

Subtracting origin coordinates from emergence coordinates,

(3/4 , 0 , 1/4) - (1/4, 1/2, 1/2) = (-1/2, 1/2, 1/4)

Multiply by 4 to convert all fractions to integers 4 x (-1/2, 1/2, 1/4) = (-2, 2, 1) Therefore, the direction indices are [ 2 2 1 ]

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

Miller Indices

• Miller Indices are are used to refer to specific lattice planes of atoms.

• They are reciprocals of the fractional intercepts (with fractions cleared) that the plane makes with the crystallographic x,y and z axes of three nonparallel edges of the cubic unit cell.

Miller Indices =(111)

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

Miller Indices - Procedure

Choose a plane that does not pass through origin

Determine the x,y and z intercepts of the plane

Find the reciprocals of the intercepts

Fractions? Clear fractions by

multiplying by an integer to determine smallest set

of whole numbers

Enclose in parenthesis (hkl)where h,k,l are miller indicesof cubic crystal plane

forx,y and z axes. Eg: (111)

Place a ‘bar’ over the Negative indices

Yes

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

Miller Indices - Examples

• Intercepts of the plane at x,y & z axes are 1, ∞ and ∞

• Taking reciprocals we get (1,0,0).

• Miller indices are (100). *******************

• Intercepts are 1/3, 2/3 & 1. • taking reciprocals we get (3,

3/2, 1). • Multiplying by 2 to clear

fractions, we get (6,3,2). • Miller indices are (632).

x x

(100)

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

Miller Indices - Examples

• Plot the plane (101) Taking reciprocals of the indices we get (1 ∞ 1). The intercepts of the plane are x=1, y= ∞ (parallel to y) and z=1.

****************************** • Plot the plane (2 2 1)

Taking reciprocals of the indices we get (1/2 1/2 1). The intercepts of the plane are x=1/2, y= 1/2 and z=1.

Figure EP3.7 a

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

Miller Indices - Example

• Plot the plane (110) The reciprocals are (1,-1, ∞) The intercepts are x=1, y=-1 and z= ∞ (parallel to z axis) To show this plane in a single unit cell, the origin is moved along the positive direction of y axis by 1 unit.

z(110)

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

Miller Indices – Important Relationship

• Direction indices of a direction perpendicular to a crystal plane are same as miller indices of the plane.

• Example:-

• Interplanar spacing between parallel closest planes with same miller indices is given by

lkh d

a hkl 222

++ =

Figure EP3.7b

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

Planes and Directions in Hexagonal Unit Cells

• Four indices are used (hkil) called as Miller- Bravais indices.

• Four axes are used (a1, a2, a3 and c). • Reciprocal of the intercepts that a crystal

plane makes with the a1, a2, a3 and c axes give the h,k,I and l indices respectively.

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

Hexagonal Unit Cell - Examples

• Basal Planes:- Intercepts a1 = ∞

a2 = ∞ a3 = ∞ c = 1 (hkli) = (0001)

• Prism Planes :- For plane ABCD,

Intercepts a1 = 1 a2 = ∞ a3 = -1 c = ∞

(hkli) = (1010)

Figure 3.17 a&b

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

Directions in HCP Unit Cells

• Indicated by 4 indices [uvtw]. • u,v,t and w are lattice vectors in a1, a2, a3 and c directions

respectively. • Example:-

For a1, a2, a3 directions, the direction indices are [ 2 1 1 0], [1 2 1 0] and [ 1 1 2 0] respectively.

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

Comparison of FCC and HCP crystals

• Both FCC and HCP are close packed and have APF 0.74.

• FCC crystal is close packed in (111) plane while HCP is close packed in (0001) plane.

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

Structural Difference between HCP and FCC

Consider a layer of atoms (Plane ‘A’)

Another layer (plane ‘B’) of atoms is placed in ‘a’

Void of plane ‘A’

Third layer of Atoms placed in ‘b’ Voids of plane ‘B’. (Identical

to plane ‘A’.) HCP crystal.

Third layer of Atoms placed in ‘a’ voids of plane ‘B’. Resulting In 3rd Plane C. FCC crystal.

Plane A ‘a’ void ‘b’ void

Plane A Plane B

‘a’ void ‘b’ void

Plane A Plane B

Plane A

Plane A Plane B

Plane C

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

Volume Density

• Volume density of metal =

• Example:- Copper (FCC) has atomic mass of 63.54 g/mol and atomic radius of 0.1278 nm.

vρ Mass/Unit cellVolume/Unit cell=

a= 2

4R = 2

1278.04 nm× = 0.361 nm

Volume of unit cell = V= a3 = (0.361nm)3 = 4.7 x 10-29 m3

vρ

FCC unit cell has 4 atoms.

Mass of unit cell = m =  





 





−

molatmos

molgatoms 6

/107.4

)/54.63)(4( = 4.22 x 10-28 Mg

33329

98.898.8 107.4

1022.4

m ==

× ==

−

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

Planar Atomic Density

• Planar atomic density=

• Example:- In Iron (BCC, a=0.287), The (100) plane intersects center of 5 atoms (Four ¼ and 1 full atom).

 Equivalent number of atoms = (4 x ¼ ) + 1 = 2 atoms Area of 110 plane =

ρ p=

Equivalent number of atoms whose centers are intersected by selected area

Selected area

222 aaa =×

ρ p ( )2287.02

2 =

1072.12.17

mmnm

atoms × ==

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

Linear Atomic Density

• Linear atomic density =

• Example:- For a FCC copper crystal (a=0.361), the [110] direction intersects 2 half diameters and 1 full diameter.

 Therefore, it intersects ½ + ½ + 1 = 2 atomic diameters.

Length of line =

ρ l

Number of atomic diameters intersected by selected length of line in direction of interest

Selected length of line

atoms

atoms 61092.392.3

361.02

2 × ==

× =ρ

nm361.02 ×

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

Polymorphism or Allotropy

• Metals exist in more than one crystalline form. This is caller polymorphism or allotropy.

• Temperature and pressure leads to change in crystalline forms.

• Example:- Iron exists in both BCC and FCC form depending on the temperature.

-2730C 9120C 13940C 15390C

α Iron BCC

γ Iron FCC

δ Iron BCC

Liquid Iron

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

Crystal Structure Analysis

• Information about crystal structure are obtained using X- Rays.

• The X-rays used are about the same wavelength (0.05- 0.25 nm) as distance between crystal lattice planes.

35 KV

(Eg: Molybdenum)

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

X-Ray Spectrum of Molybdenum

• X-Ray spectrum of Molybdenum is obtained when Molybdenum is used as target metal.

• Kα and Kβ are characteristic of an element.

• For Molybdenum Kα occurs at wave length of about 0.07nm.

• Electrons of n=1 shell of target metal are knocked out by bombarding electrons.

• Electrons of higher level drop down by releasing energy to replace lost electrons

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

X-Ray Diffraction

• Crystal planes of target metal act as mirrors reflecting X- ray beam.

• If rays leaving a set of planes are out of phase (as in case of arbitrary angle of incidence) no reinforced beam is produced.

• If rays leaving are in phase, reinforced beams are produced.

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

X-Ray Diffraction (Cont..)

• For rays reflected from different planes to be in phase, the extra distance traveled by a ray should be a integral multiple of wave length λ .

nλ = MP + PN (n = 1,2…)

n is order of diffraction

If dhkl is interplanar distance,

Then MP = PN = dhkl.Sinθ

Therefore, λ = 2 dhkl.Sinθ

After A.G. Guy and J.J. Hren, “Elements of Physical Metallurgy,” 3d ed., Addison-Wesley, 1974, p.201.)

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

Interpreting Diffraction Data

• We know that

2222 2

222

lkh Sin

lkh

aSin

dSin

lkh

a d hkl

++ =

λ θ

θ λ

θλSince

Note that the wavelength λ and lattice constant a are the same For both incoming and outgoing radiation.

Substituting for d,

Therefore

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

Interpreting Diffraction Data (Cont..)

• For planes ‘A’ and ‘B’ we get two equations

2222 2

)(

lkh Sin

BBB B

AAA A

++ =

λ θ

λ θ (For plane ‘A’)

(For plane ‘B’)

Dividing each other, we get

)(

)( 222

222

BBB

AAA

lkh

Sin

++ =

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

X-Ray Diffraction Analysis

• Powdered specimen is used for X-ray diffraction analysis as the random orientation facilitates different angle of incidence.

• Radiation counter detects angle and intensity of diffracted beam.

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

Diffraction Condition for Cubic Cells

• For BCC structure, diffraction occurs only on planes whose miller indices when added together total to an even number.

I.e. (h+k+l) = even Reflections present (h+k+l) = odd Reflections absent

• For FCC structure, diffraction occurs only on planes whose miller indices are either all even or all odd. I.e. (h,k,l) all even Reflections present

(h,k,l) all odd Reflections present (h,k,l) not all even or all odd Reflections absent.

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

Interpreting Experimental Data

• For BCC crystals, the first two sets of diffracting planes are {110} and {200} planes.

Therefore

• For FCC crystals the first two sets of diffracting planes are {111} and {200} planes

Therefore

5.0 )002(

)011( 222

222

= ++

++ =

Sin

75.0 )002(

)111( 222

222

= ++

++ =

Sin

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

Crystal Structure of Unknown Metal

Unknown metal

Crystallographic Analysis

FCC Crystal

Structure

BCC Crystal

Structure

75.02 2

= B

Sin Sin

θ θ 5.0

= B

Sin

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

Amorphous Materials

• Random spatial positions of atoms • Polymers: Secondary bonds do not allow

formation of parallel and tightly packed chains during solidification.  Polymers can be semicrystalline.

• Glass is a ceramic made up of SiO4 4- tetrahedron subunits – limited mobility.

• Rapid cooling of metals (10 8 K/s) can give rise to amorphous structure (metallic glass).

• Metallic glass has superior metallic properties.

Definitions • Crystal structure: the manner in which atoms, ions and molecules are spatially arranged • Lattice: a three dimensional array of points coinciding with atom positions • Allotropy: a property of an element to coexist in more than one crystal structure • Amorphous materials: materials like glasses that have no long range order or crystal structure • Isotropic: having same properties in all directions • Anisotropic: property of materials that have different properties in different directions • Unit cell: subdivision of a crystal structure into small repeat entities • Basis or motif: a group of atoms associated with a lattice point • Coordination number: number of nearest neighbors of an atom in a crystal structure • Atomic packing fraction: fraction of plane, volume or direction that is actually occupied by atoms • Lattice parameter: the unit cell edge lengths and angle between them • Closed packed directions: directions in a crystal lattice along which atoms are in contact

Definitions • Cubic site: an interstitial site with coordination number of eight • Tetrahedral site: an interstitial site with coordination number of four • Octahedral site: an interstitial site with coordination number of six • Bravais lattice: fourteen possible lattice arrangements that are possible using lattice points in 3D

space • Miller indices: a notation used to describe specific crystallographic planes and directions • Miller indices of directions: are vector components of the directions resolved along each of the

coordinate axes • Miller indices of planes: are defined as reciprocals of fractional intercepts that the planes makes

with coordinate axes • Metallic glass: metals with an amorphous atomic structure • Linear atomic density: the number of lattice points per unit length along a direction • Planar density: the number of atoms whose centers are intersected by the plane per unit area • Volume density: the ratio of mass per unit volume of a unit cell

Ch03-revised-D2L

CHAPTER �3
The Space Lattice and Unit Cells
Building blocks
Unit cell
Crystal Systems and Bravais Lattice
Types of Unit Cells
Types of Unit Cells (Cont..)
Types of Unit Cells (Cont..)
Principal Metallic Crystal Structures
Body Centered Cubic (BCC) Crystal Structure
BCC Crystal Structure (Cont..)
Atomic Packing Factor of BCC Structure
Face Centered Cubic (FCC) Crystal Structure
FCC Crystal Structure (Cont..)
Hexagonal Close-Packed Structure
HCP Crystal Structure (Cont..)
YouTube videos links
Atom Positions in Cubic Unit Cells
Directions in Cubic Unit Cells
Procedure to Find Direction Indices
Direction Indices - Example
Miller Indices
Miller Indices - Procedure
Miller Indices - Examples
Miller Indices - Examples
Miller Indices - Example
Miller Indices – Important Relationship
Planes and Directions in Hexagonal Unit Cells
Hexagonal Unit Cell - Examples
Directions in HCP Unit Cells
Comparison of FCC and HCP crystals
Structural Difference between HCP and FCC
Volume Density
Planar Atomic Density
Linear Atomic Density
Polymorphism or Allotropy
Crystal Structure Analysis
X-Ray Spectrum of Molybdenum
X-Ray Diffraction
X-Ray Diffraction (Cont..)
Interpreting Diffraction Data
Interpreting Diffraction Data (Cont..)
X-Ray Diffraction Analysis
Diffraction Condition for Cubic Cells
Interpreting Experimental Data
Crystal Structure of Unknown Metal
Amorphous Materials

Definitions-ch-3

Definitions
Definitions