lectures 2 to 5 - crystallography

7/23/2019 Lectures 2 to 5 - Crystallography

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ISSUES TO ADDRESS...

How do atoms assemble into solid structures?

(for now, focus on metals)

its structure?

When do material properties vary with the

sample (i.e., part) orientation?

1

Energy and Packing

Non dense, random packingEnergy

typical neighbor

r

bond length

typical neighbor

Dense, ordered packing

on energy

Energy

typical neighborbond length

rtypical neighborbond energy

2

,

Materials and Packing

atoms pack in periodic, 3D arraysCrystalline materials...

-metals

-many ceramics

-some ol mers cr stalline SiO2

typical of:

Adapted from Fig. 3.22 (a), Callister 7e.

Si Oxygen

atoms have no periodic packing

oncrys a ne ma er a s...

-complex structures occurs for:

-rapid cooling

noncrystalline SiO2"Amorphous" = Noncrystalline

3

Adapted from Fig. 3.22(b ), Callister 7e.

Crystal Structure

Space lattice: Three dimensional network of imagina

lines connectin the atoms is called the s ace latti

(A three dimensional pattern of points in space).

4

complete lattice pattern of a crystal (The smallest unit having

the full symmetry of the crystal).

7 crystal systems

The specific unit cell for metal

The edges a, b, and c; and , and

5Fig. 3.4, Callister 7e.

Crystal Systems

6


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Metallic Crystal Structures

How can we stack metal atoms to minimize

em t s ace?

2-dimensions

vs.

7

ow s ac ese - ayers o ma e - s ruc ures

Metallic Crystal Structures

Tend to be densely packed.

- 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...

8

Simple Cubic Structure (SC)

Rare due to low packing denisty (only Po has this structure)

Close-packed directions are cube edges.

Coordination # = 6

(# nearest neighbors)

9(Courtesy P.M. Anderson)

Atomic Packing Factor (APF)

APF =Volume of atoms in unit cell*

Volume of unit cell

*assume hard spheres

.

atomsvolume

APF =

4

3 (0.5a) 31unit cella

R=0.5a

a3

unit cell

volume

close-packed directionscontains 8 x 1/8 =

10

Adapted from Fig. 3.23,

Callister 7e.

1 atom/unit cell

Body Centered Cubic Structure (BCC)

Atoms touch each other along cube diagonals.--Note: All atoms are identical; the center atom is shaded

=

eren y ony or ease o v ewng.

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

Adapted from Fig. 3.2,

Callister 7e.

11(Courtesy P.M. Anderson)

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

Atomic Packing Factor: BCC APF for a body-centered cubic structure = 0.68

a3

a

-

a2

length = 4R=

3 aa

RAdapted from

Fig. 3.2(a), Callister 7e.

APF =

4

3 ( 3a/4)32unit cell atom

vo ume

12

a3

unit cell

volume


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Crystallographic Directions

1. Vector repositioned (if necessary) to pass

z Algorithm

hrough origin.

2. Read off projections in terms of

unit cell dimensions a, b, and c

.

4. Enclose in square brackets, no commas

[uvw]x

y

ex: 1, 0, => 2, 0, 1 => [ 201]

-1, 1, 1 where overbar represents a

negative index

[111]=>

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families of directions

Linear Density

Linear Density of Atoms LD =Unit length of direction vec

Number of atoms

ex: linear density of Al in [110][110]

a = 0.405 nm

a# atoms

2

length

3.5 nma2

LD

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HCP Crystallographic Directionsz

Algorithm

. ec or repos one necessary o pass

through origin.

2. Read off projections in terms of unit

1, 2, 3,

3. Adjust to smallest integer values

4. Enclose in square brackets, no commas-a3

2

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

a2

-a32

a2

a1

[1120]ex: , , -1, 0 =>

dashed red lines indicate

a3

2

a1

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projections onto a1 and a2 axes a1

HCP Crystallographic Directions Hexagonal Crystals

4 parameter Miller-Bravais lattice coordinates are

' ' 're a e o e rec on n ces .e., u v w as

follows.z

u )'v'u2(1

-

]uvtw[]'w'v'u[

v )'u'v2(1

-a2

'ww

t )vu( +--

a3

a1

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Fig. 3.8(a), Callister 7e.

Crystallographic Planes

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Adapted from Fig. 3.9, Callister 7e.

Crystallographic Planes

Miller Indices: Reciprocals of the (three) axialintercepts for a plane, cleared of fractions &common multiples. All parallel planes have

same Miller indices.

Algorithm.

terms of a, b, c2. Take reciprocals of intercepts

.4. Enclose in parentheses, no

commas i.e., (hkl)

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Planar Density of (111) IronSolution (cont): (111) plane 1 atom in plane/ unit surface cell

atoms in planea2atoms above plane

atoms below plane

ah23

333

2

2

R3

16R

3

42a3ah2area

atoms

1

= =nm2

atoms7.0

m2atoms

0.70 x 1019

316Planar Density =

2D repeat unit

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R32D repeat unit

X-Ray Diffraction

Diffraction ratin s must have s acin s com arable tothe wavelength of diffracted radiation.

Cant resolve spacings

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atoms.

X-Rays to Determine Crystal Structure Incoming X-rays diffract from crystal planes.

Adapted from Fig. 3.19 ,

reflections mustbe in phase fora detectable signal

extradistance

Callister 7e.

spacingbetween

planes

d

travelledby wave 2

X-rayintensit

nMeasurement of

(from

detector)

2 sinc , c,

allows computation of

planar spacing, d.

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c

X-Ray Diffraction Patternz

c

zc

z

c

(110)

x

a b

tive)

x

ya b

x

ya b

ensity(rela

Int

Diffraction angle 2

Diffraction pattern for polycrystalline -iron (BCC)

40

Adapted from Fig. 3.20, Callister 5e.

SUMMARY

Atoms may assemble into crystalline or amorphous

.

Common metallic crystal structures are FCC, BCC, and

HCP. Coordination numberand atomic packing factor

We can predict the density of a material, provided we

are the same for both FCC and HCP crystal structures.

, ,

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.

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SUMMARY

Materials can be single crystals orpolycrystalline.

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

Some materials can have more than one crystal

with randomly oriented grains.

s ruc ure. s s re erre o as po ymorp sm or

allotropy).

-

interplanar spacing determinations.

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lectures 2 to 5 - crystallography

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