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Crystallography and Structure. ME 2105 R. R. Lindeke. Overview:. Crystal Structure – matter assumes a periodic shape Non-Crystalline or Amorphous “structures” no long range periodic shapes FCC, BCC and HCP – common for metals Xtal Systems – not structures but potentials

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crystallography and structure

Crystallography and Structure

ME 2105

R. R. Lindeke

overview
Overview:
  • Crystal Structure – matter assumes a periodic shape
    • Non-Crystalline or Amorphous “structures” no long range periodic shapes
    • FCC, BCC and HCP – common for metals
    • Xtal Systems – not structures but potentials
  • Point, Direction and Planer ID’ing in Xtals
  • X-Ray Diffraction and Xtal Structure
energy and packing
Energy and Packing

Energy

typical neighbor

bond length

typical neighbor

r

bond energy

• Dense, ordered packing

Energy

typical neighbor

bond length

r

typical neighbor

bond energy

• Non dense, random packing

Dense, ordered packed structures tend to have

lower energies.

crystal structures
CRYSTAL STRUCTURES

Crystal Structure

  • Means: PERIODIC ARRANGEMENT OF ATOMS/IONS
  • OVER LARGE ATOMIC DISTANCES
  • Leads to structure displayingLONG-RANGE ORDER that is

Measurable and Quantifiable

All metals, many ceramics, some polymers exhibit this “High Bond Energy” – More Closely Packed Structure

materials lacking long range order
Materials Lacking Long range order

Amorphous Materials

These less densely packed lower bond energy “structures” can be found in Metal are observed in Ceramic GLASS and many “plastics”

crystal systems some definitional information

Fig. 3.4, Callister 7e.

Crystal Systems – Some Definitional information

Unit cell: smallest repetitive volume which contains the complete lattice pattern of a crystal.

7 crystal systems of varying symmetry are known

These systems are built by changing the lattice parameters:a, b, and c are the edge lengths

, , and  are interaxial angles

slide7

Crystal Systems

Crystal structures are divided into groups according to unit cell geometry (symmetry).

metallic crystal structures
Metallic Crystal Structures

• Tend to be densely packed.

• Reasons for dense packing:

- Typically, only one element is present, so all atomic

radii are the same.

- Metallic bonding is not directional.

- Nearest neighbor distances tend to be small in

order to lower bond energy.

- Electron cloud shields cores from each other

• Have the simplest crystal structures.

We will examine three such structures (those of engineering importance) called: FCC, BCC and HCP – with a nod to Simple Cubic

simple cubic structure sc
Simple Cubic Structure (SC)

• Rare due to low packing density (only Po – Polonium -- has this structure)

• Close-packed directions are cube edges.

• Coordination No. = 6

(# nearest neighbors) for each atom as seen

(Courtesy P.M. Anderson)

atomic packing factor apf
Atomic Packing Factor (APF)

volume

atoms

atom

4

a

3

unit cell

p

(0.5a)

3

R=0.5a

volume

close-packed directions

unit cell

contains (8 x 1/8) =

1

atom/unit cell

Adapted from Fig. 3.23,

Callister 7e.

Volume of atoms in unit cell*

APF =

Volume of unit cell

*assume hard spheres

• APF for a simple cubic structure = 0.52

1

APF =

3

a

Here: a = Rat*2

Where Rat is the ‘handbook’ atomic radius

body centered cubic structure bcc
Body Centered Cubic Structure (BCC)

• Atoms touch each other along cube diagonals.

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

differently only for ease of viewing.

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

• Coordination # = 8

Adapted from Fig. 3.2,

Callister 7e.

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

(Courtesy P.M. Anderson)

atomic packing factor bcc
Atomic Packing Factor: BCC

a

3

a

2

Close-packed directions:

R

3 a

length = 4R =

a

atoms

volume

4

3

p

(

3

a/4

)

2

unit cell

atom

3

APF =

volume

3

a

unit cell

a

Adapted from

Fig. 3.2(a), Callister 7e.

• APF for a body-centered cubic structure = 0.68

face centered cubic structure fcc
Face Centered Cubic Structure (FCC)

• Atoms touch each other along face diagonals.

--Note: All atoms are identical; the face-centered atoms are shaded

differently only for ease of viewing.

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

• Coordination # = 12

Adapted from Fig. 3.1, Callister 7e.

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

(Courtesy P.M. Anderson)

atomic packing factor fcc
Atomic Packing Factor: FCC

Close-packed directions:

2 a

length = 4R =

2 a

Unit cell contains:

6 x1/2 + 8 x1/8

=

4 atoms/unit cell

a

atoms

volume

4

3

p

(

2

a/4

)

4

unit cell

atom

3

APF =

volume

3

a

unit cell

• APF for a face-centered cubic structure = 0.74

The maximum achievable APF!

(a = 22*R)

Adapted from

Fig. 3.1(a),

Callister 7e.

hexagonal close packed structure hcp
Hexagonal Close-Packed Structure (HCP)

A sites

Top

layer

c

Middle layer

B

sites

A sites

Bottom layer

a

ex: Cd, Mg, Ti, Zn

• ABAB... Stacking Sequence

• 3D Projection

• 2D Projection

Adapted from Fig. 3.3(a),

Callister 7e.

6 atoms/unit cell

• Coordination # = 12

• APF = 0.74

• c/a = 1.633 (ideal)

slide18

We find that both FCC & HCP are highest density packing schemes (APF = .74) – this illustration shows their differences as the closest packed planes are “built-up”

theoretical density r

nA

VCNA

Mass

of

Atoms

in

Unit

Cell

Total

Volume

of

Unit

Cell

 =

Theoretical Density, r

Density =  =

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

theoretical density r20

R

a

2

52.00

theoretical

atoms

= 7.18 g/cm3

g

 =

unit cell

ractual

= 7.19 g/cm3

mol

atoms

3

a

6.023x1023

mol

volume

unit cell

Theoretical Density, r
  • Ex: Cr (BCC)

A =52.00 g/mol

R = 0.125 nm

n = 2

 a = 4R/3 = 0.2887 nm

slide22

Cesium chloride (CsCl) unit cell showing (a) ion positions and the two ions per lattice point and (b) full-size ions. Note that the Cs+−Cl− pair associated with a given lattice point is not a molecule because the ionic bonding is nondirectional and because a given Cs+is equally bonded to eight adjacent Cl−, and vice versa. [Part (b) courtesy of Accelrys, Inc.]

slide23

Sodium chloride (NaCl) structure showing (a) ion positions in a unit cell, (b) full-size ions, and (c) many adjacent unit cells. [Parts (b) and (c) courtesy of Accelrys, Inc.]

slide24
Fluorite (CaF2) unit cell showing (a) ion positions and (b) full-size ions. [Part (b) courtesy of Accelrys, Inc.]
polymorphism also in metals

iron system:

liquid

1538ºC

-Fe

BCC

1394ºC

-Fe

FCC

912ºC

BCC

-Fe

Polymorphism: Also in Metals
  • Two or more distinct crystal structures for the same material (allotropy/polymorphism)  titanium

 (HCP), (BCC)-Ti

carbon:

diamond, graphite

slide28

The corundum (Al2O3) unit cell is shown superimposed on the repeated stacking of layers of close-packed O2−ions. The Al3+ions fill two-thirds of the small (octahedral) interstices between adjacent layers.

slide29

Exploded view of the kaolinite unit cell, 2(OH)4Al2Si2O5. (From F. H. Norton, Elements of Ceramics, 2nd ed., Addison-Wesley Publishing Co., Inc., Reading, MA, 1974.)

slide30

Transmission electron micrograph of the structure of clay platelets. This microscopic-scale structure is a manifestation of the layered crystal structure shown in the previous slide. (Courtesy of I. A. Aksay.)

slide31

(a) An exploded view of the graphite (C) unit cell. (From F. H. Norton, Elements of Ceramics, 2nd ed., Addison-Wesley Publishing Co., Inc., Reading, MA, 1974.) (b) A schematic of the nature of graphite’s layered structure. (From W. D. Kingery, H. K. Bowen, and D. R. Uhlmann, Introduction to Ceramics, 2nd ed., John Wiley & Sons, Inc., NY, 1976.)

slide32
(a) C60molecule, or buckyball. (b) Cylindrical array of hexagonal rings of carbon atoms, or buckytube. (Courtesy of Accelrys, Inc.)
slide33

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. (Courtesy of Accelrys, Inc.)

slide34

Weaving-like pattern of folded polymeric chains that occurs in thin crystal platelets of polyethylene. (From D. J. Williams, Polymer Science and Engineering, Prentice Hall, Inc., Englewood Cliffs, NJ, 1971.)

slide35

Diamond cubic unit cell showing (a) atom positions. There are two atoms per lattice point (note the outlined example). Each atom is tetrahedrally coordinated. (b) The actual packing of full-size atoms associated with the unit cell. [Part (b) courtesy of Accelrys, Inc.]

slide36

Zinc blende (ZnS) unit cell showing (a) ion positions. There are two ions per lattice point (note the outlined example). Compare this structure with the diamond cubic structure (Figure 3.20a). (b) The actual packing of full-size ions associated with the unit cell. [Part (b) courtesy of Accelrys, Inc.]

densities of material classes

r

r

r

metals

ceramics

polymers

Platinum

Gold, W

Tantalum

Silver, Mo

Cu,Ni

Steels

Tin, Zinc

Zirconia

Titanium

Al oxide

Diamond

Si nitride

Aluminum

Glass

-

soda

Glass fibers

Concrete

PTFE

Silicon

GFRE*

Carbon

fibers

Magnesium

G

raphite

CFRE

*

Silicone

A

ramid fibers

PVC

AFRE

*

PET

PC

H

DPE, PS

PP, LDPE

Wood

Densities of Material Classes

In general

Graphite/

Metals/

Composites/

Ceramics/

Polymers

>

>

Alloys

fibers

Semicond

30

Why?

2

0

*GFRE, CFRE, & AFRE are Glass,

Metals have...

• close-packing

(metallic bonding)

• often large atomic masses

Carbon, & Aramid Fiber-Reinforced

Epoxy composites (values based on

60% volume fraction of aligned fibers

10

in an epoxy matrix).

Ceramics have...

• less dense packing

• often lighter elements

5

3

4

(g/cm )

3

r

2

Polymers have...

• low packing density

(often amorphous)

• lighter elements (C,H,O)

1

0.5

Composites have...

• intermediate values

0.4

0.3

Data from Table B1, Callister 7e.

crystals as building blocks
Crystals as Building Blocks

• Some engineering applications require single crystals:

--diamond single

crystals for abrasives

--turbine blades

Fig. 8.33(c), Callister 7e.

(Fig. 8.33(c) courtesy

of Pratt and Whitney).

(Courtesy Martin Deakins,

GE Superabrasives, Worthington, OH. Used with permission.)

• Properties of crystalline materials

often related to crystal structure.

--Ex: Quartz fractures more easily along some crystal planes than others.

(Courtesy P.M. Anderson)

polycrystals
Polycrystals

Anisotropic

• Most engineering materials are polycrystals.

Adapted from Fig. K, color inset pages of Callister 5e.

(Fig. K is courtesy of Paul E. Danielson, Teledyne Wah Chang Albany)

1 mm

Isotropic

• Nb-Hf-W plate with an electron beam weld.

• Each "grain" is a single crystal.

• If grains are randomly oriented,

overall component properties are not directional.

• Grain sizes typ. range from 1 nm to 2 cm

(i.e., from a few to millions of atomic layers).

single vs polycrystals
Single vs Polycrystals

E (diagonal) = 273 GPa

E (edge) = 125 GPa

• Single Crystals

Data from Table 3.3, Callister 7e.

(Source of data is R.W. Hertzberg, Deformation and Fracture Mechanics of Engineering Materials, 3rd ed., John Wiley and Sons, 1989.)

-Properties vary with

direction: anisotropic.

-Example: the modulus

of elasticity (E) in BCC iron:

• Polycrystals

200 mm

-Properties may/may not

vary with direction.

-If grains are randomly

oriented: isotropic.

(Epoly iron = 210 GPa)

-If grains are textured,

anisotropic.

Adapted from Fig. 4.14(b), Callister 7e.

(Fig. 4.14(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].)

locations in lattices point coordinates

z

111

c

y

000

b

a

x

Locations in Lattices: Point Coordinates

Point coordinates for unit cell center are

a/2, b/2, c/2 ½½½

Point coordinates for unit cell (body diagonal) corner are 111

Translation: integer multiple of lattice constants  identical position in another unit cell

z

2c

y

b

b

crystallographic directions

[111]

where ‘overbar’ represents a negative index

=>

Crystallographic Directions

Algorithm

z

1. Vector is repositioned (if necessary) to pass through the Unit Cell origin.2. Read off line projections (to principal axes of U.C.) in terms of unit cell dimensions a, b, and c3. Adjust to smallest integer values4. Enclose in square brackets, no commas

[uvw]

y

x

ex:1, 0, ½

=> 2, 0, 1

=> [201]

-1, 1, 1

families of directions <uvw>

slide44

xy z

Projections:

Projections in terms of a,b and c:

Reduction:

Enclosure [brackets]

What is this Direction ?????

0c

a/2

b

0

1

1/2

2

0

1

[120]

linear density considers equivalance and is important in slip

[110]

# atoms

2

a

-

=

=

1

LD

3.5 nm

2

a

length

Linear Density – considers equivalance and is important in Slip

Number of atoms

Unit length of direction vector

  • Linear Density of Atoms  LD = 

ex: linear density of Al in [110] direction a = 0.405 nm

# atoms CENTERED on the direction of interest!

Length is of the direction of interest within the Unit Cell

determining angles between crystallographic direction
Determining Angles Between Crystallographic Direction:

Where ui’s , vi’s & wi’s are the “Miller Indices” of the directions in question

– also (for information) If a direction has the same Miller Indicies as a plane, it is NORMAL to that plane

hcp crystallographic directions

z

a2

-

a2

a3

-a3

a

2

a1

2

a3

ex: ½, ½, -1, 0 =>

[1120]

a

1

2

dashed red lines indicate

projections onto a1 and a2 axes

a1

HCP Crystallographic Directions

Algorithm

1. Vector repositioned (if necessary) to pass

through origin.2. Read off projections in terms of unit

cell dimensions a1, a2, a3, or c3. Adjust to smallest integer values4. Enclose in square brackets, no commas

[uvtw]

Adapted from Fig. 3.8(a), Callister 7e.

hcp crystallographic directions48

z

®

u

[

'

v

'

w

'

]

[

uvtw

]

1

=

-

u

(

2

u

'

v

'

)

3

a3

a2

1

=

-

v

(

2

v

'

u

'

)

a1

3

-

=

-

+

t

(

u

v

)

=

w

w

'

Fig. 3.8(a), Callister 7e.

HCP Crystallographic Directions
  • Hexagonal Crystals
    • 4 parameter Miller-Bravais lattice coordinates are related to the direction indices (i.e., u'v'w') in the ‘3 space’ Bravais lattice as follows.
computing hcp miller bravais directional indices an alternative way

z

a3

a2

a1

-

Fig. 3.8(a), Callister 7e.

Computing HCP Miller- Bravais Directional Indices (an alternative way):

We confine ourselves to the bravais parallelopiped in the hexagon: a1-a2-Z and determine: (u’,v’w’)

Here: [1 1 0] - so now apply the models to create M-B Indices

defining crystallographic planes
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 (cubic lattices 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}

crystallographic planes families
Crystallographic Planes -- families

Adapted from Fig. 3.9, Callister 7e.

crystallographic planes

example

example

abc

abc

z

1 1 

1. Intercepts

c

1/1 1/1 1/

2. Reciprocals

1 1 0

3. Reduction

1 1 0

y

b

a

z

x

1/2  

1. Intercepts

c

1/½ 1/ 1/

2. Reciprocals

2 0 0

3. Reduction

2 0 0

y

b

a

x

Crystallographic Planes

4. Miller Indices (110)

4. Miller Indices (100)

crystallographic planes53

z

c

1/2 1 3/4

1. Intercepts

1/½ 1/1 1/¾

2. Reciprocals

2 1 4/3

y

b

a

3. Reduction

6 3 4

x

Family of Planes {hkl}

Ex: {100} = (100),

(010),

(001),

(100),

(001)

(010),

Crystallographic Planes

example

a b c

4. Miller Indices (634)

slide54

(012)

Determine the Miller indices for the plane shown in the accompanying sketch (a)

xyz

Intercepts

Intercept in terms of lattice parameters

Reciprocals

Reductions

Enclosure

a -b c/2

 -1 1/2

0 -1 2

N/A

crystallographic planes hcp

z

example

a1a2a3 c

1 

1. Intercepts

-1

1

1 1/

2. Reciprocals

-1

1

1 0

-1

1

a2

3. Reduction

1 0

-1

1

a3

4. Miller-Bravais Indices

(1011)

a1

Crystallographic Planes (HCP)
  • In hexagonal unit cells the same idea is used

Adapted from Fig. 3.8(a), Callister 7e.

crystallographic planes56
Crystallographic Planes
  • We want to examine the atomic packing of crystallographic planes – those with the same packing are equivalent and part of families
  • Iron foil can be used as a catalyst. The atomic packing of the exposed planes is important.
    • Draw (100) and (111) crystallographic planes

for Fe.

b) Calculate the planar density for each of these planes.

planar density of 100 iron

4

3

=

a

R

3

atoms

2

2D repeat unit

1

a

1

atoms

atoms

12.1

= 1.2 x 1019

=

=

Planar Density =

2

nm2

m2

4

3

R

3

area

2D repeat unit

Planar Density of (100) Iron

2D repeat unit

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

(100)

Radius of iron R = 0.1241 nm

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

Atoms: wholly contained and centered in/on plane within U.C., area of plane in U.C.

planar density of 111 iron

atoms in plane

atoms above plane

atoms below plane

atoms

2D repeat unit

3*1/6

atoms

Planar Density =

atoms

=

=

7.0

0.70 x 1019

m2

nm2

area

2D repeat unit

Planar Density of (111) Iron

1/2 atom centered on plane/ unit cell

a

2

Solution (cont):  (111) plane

2D repeat unit

3

=

h

a

2

Area 2D Unit: ½ hb = ½*[(3/2)a][(2)a]=1/2(3)a2=8R2/(3)

section 3 16 x ray diffraction
Section 3.16 - X-Ray Diffraction
  • Diffraction gratings must have spacings comparable to the wavelength of diffracted radiation.
  • Can’t resolve spacings 
  • Spacing is the distance between parallel planes of atoms.  
figure 3 32 relationship of the bragg angle and the experimentally measured diffraction angle 2

X-ray

nl

intensity

q

d=

c

2sin

(from

detector)

q

q

c

Figure 3.32Relationship of the Bragg angle (θ) and the experimentally measured diffraction angle (2θ).
x rays to determine crystal structure

detector

“1”

incoming

reflections must

X-rays

“2”

be in phase for

“1”

a detectable signal!

outgoing X-rays

l

extra

“2”

Adapted from Fig. 3.19, Callister 7e.

distance

q

q

traveled

spacing

by wave “2”

d

between

planes

X-ray

nl

intensity

q

d=

c

2sin

(from

detector)

q

q

c

X-Rays to Determine Crystal Structure

• Incoming X-rays diffract from crystal planes.

Measurement of critical angle, qc, allows computation of planar spacing, d.

For Cubic Crystals:

h, k, l are Miller Indices

figure 3 34 a an x ray diffractometer courtesy of scintag inc b a schematic of the experiment
Figure 3.34(a) An x-ray diffractometer. (Courtesy of Scintag, Inc.) (b) A schematic of the experiment.
x ray diffraction pattern

z

z

z

c

c

c

y

y

y

a

a

a

b

b

b

x

x

x

X-Ray Diffraction Pattern

(110)

(211)

Intensity (relative)

(200)

Diffraction angle 2q

Diffraction pattern for polycrystalline a-iron (BCC)

Adapted from Fig. 3.20, Callister 5e.

summary
SUMMARY

• Atoms may assemble into crystalline or

amorphous structures.

• Common metallic crystal structures are FCC, BCC, and

HCP. Coordination number and atomic packing factor

are the same for both FCC and HCP crystal structures.

• We can predict the density of a material, provided we

know the atomic weight, atomic radius, and crystal

geometry (e.g., FCC, BCC, HCP).

• Crystallographic points, directions and planes are

specified in terms of indexing schemes.

Crystallographic directions and planes are related

to atomic linear densities and planar densities.

summary66
SUMMARY

• Materials can be single crystals or polycrystalline.

Material properties generally vary with single crystal

orientation (i.e., they are anisotropic), but are generally

non-directional (i.e., they are isotropic) in polycrystals

with randomly oriented grains.

• Some materials can have more than one crystal

structure. This is referred to as polymorphism (or

allotropy).

• X-ray diffraction is used for crystal structure and

interplanar spacing determinations.