Chapter 03 Annot

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• Crystalline – repetitive 3-D structure

- all metals, many ceramics andsome polymers

• Atomic Hard Sphere Model

represent atoms as solid spheres touching

each other.

• Lattice

3-D array of points coinciding with atom

positions

CHAPTER 3: STRUCTURE OF

CRYSTALLINE SOLIDS• 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.

• Have the simplest crystal structures.

Types: 1) FCC 2) BCC 3) HCP

Focus on the unit cell: smallest repetitive entity

Properties :•Coordination number – number of nearest neighbor or

touching atoms

•Atomic Packing Factor (APF) – ratio of volume of atoms in

unit cell to the total unit cell volume

METALLIC CRYSTAL STRUCTURE

• Coordination # = 12

• Atom at each corner, and at center of each face.--Note: All atoms are identical; the face-centered atoms are shaded

differently only for ease of viewing.

FACE CENTERED CUBIC

STRUCTURE (FCC)

a

• APF for a body-centered cubic structure = 0.74

ATOMIC PACKING FACTOR: FCC


• Atom at each corner, and at the center of cube.--Note: All atoms are identical; the center atom is shaded

differently only for ease of viewing.

BODY CENTERED CUBIC

STRUCTURE (BCC)

aR

• APF for a body-centered cubic structure = 0.68

ATOMIC PACKING FACTOR: BCC




• ABAB... Stacking Sequence

• 3D Projection • 2D Projection

A sites

B sites

A sites Bottom layer

Middle layer

Top layer

HEXAGONAL CLOSE-PACKED

STRUCTURE (HCP)

• APF = 0.74

• ABCABC... Stacking Sequence

• 2D Projection

A sites

B sites

C sites

B B

B

BB

B BC C

CA

A

• FCC Unit CellA

BC

FCC STACKING SEQUENCE

HCP APF

• APF = 0.74

aac

Ra

633.13

22

2

≈=

=

ρ = n A

VcNA

# 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

Example: Density Computation

Example: Compute the theoretical density of Copper

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)

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

Result: theoretical ρCu = 8.89 g/cm3

Compare to actual: ρCu = 8.94 g/cm3

-7

Element AluminumArgonBariumBerylliumBoronBromineCadmium

CalciumCarbonCesiumChlorineChromiumCobalt CopperFlourineGalliumGermaniumGoldHeliumHydrogen

SymbolAlArBaBeBBrCd

CaCCsClCrCoCu FGaGeAuHeH

At. Weight (amu)26.9839.95137.339.01210.8179.90112.41

40.0812.011132.9135.4552.0058.9363.55 19.0069.7272.59196.974.0031.008

Atomic radius(nm)0.143------0.2170.114------------0.149

0.1970.0710.265------0.1250.1250.128 ------0.1220.1220.144------------

Density

(g/cm3)2.71------3.51.852.34------8.65

1.552.251.87------7.198.98.94 ------5.905.3219.32------------

CrystalStructureFCC------BCCHCPRhomb------HCP

FCCHexBCC------BCCHCPFCC ------Ortho.Dia. cubicFCC------------

Adapted from

Table, "Charac-teristics of

SelectedElements",

inside front

cover,

Callister 6e.

Characteristics of Selected Elements at 20C



Polymorphism and AllotropyPolymorphism

-Having more than one crystal structure (change with T & P)

-Ex. Low-Carbon Steel can exist as BCC or FCC

Allotropy

-Polymorphism in elemental solid

-Ex. Carbon can exist as graphite or diamond

Temperature, C

BCC Stable

FCC Stable

914

1391

1536

shorter

longer!shorter!

longer

Tc 768 magnet falls off

BCC Stable

Liquid

heat up

cool down

• Atoms may have crystalline or amorphous

structures.

• We can predict the density of a material from the

atomic weight , atomic radius, and crystal

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

• Anisotropic material - properties vary with

direction – true for a for a single crystal

orientation.

• Isotropic material - properties are non-

directional – true for polycrystals with randomly

oriented grains.

SUMMARY

Crystal Systems

Unit Cell:

Parameters:

γ β ,,,,, cba

7 Crystal Systems:

-Cubic

-Hexagonal

-Tetragonal

-Orthorhombic

-Rhombohedral

-Monoclinic

-Triclini

Crystallographic Positions (Coordinates)

- Coordinates of points expressed as fractions

of unit cell dimensions

Crystallographic Directions

-defined by line between two points

Steps:1. Translate vector to pass at

origin.

2. Determine lengths of

components (projections) in

terms of unit cell dimensions.3. Multiply or divide by a common

factor to reduce to smallest

possible integer values, uvw.

4. Direction: [uvw]

Family of Directions – nonparallel directions which are

equivalent (same atom spacing).

Ex. Cubic: <100> family – [100], [010], [001]

Tetragonal <100> - [100], [010]

Crystallographic Planes (Miller Indices)

Steps:

1. If the plane passes through origin, translate

origin.

2. Determine lengths of planar intercepts in

terms of unit cell dimensions.

3. Take reciprocals of the numbers in Step 2.

4. Multiply or divide by a common factor to

reduce to integers: hkl

5. Enclose: (hkl)

Example:



Planes for Hexagonal Crystals

Four axes: z,,, 321 α α α

Example:

Linear Density

– fraction of the length of the line intersected by atoms.

Planar Density

- Fraction of the area occupied by atoms (count

atoms only if plane intersect its center.

Example: For simple cubic crystal,

find:

1. Linear density for [111] direction,

2. Planar density for (100) plane.

d=nλ/2sinθc

x-rayintensity(fromdetector)

θ

θc

• Incoming X-rays diffract from crystal planes.

• Measurement of:Critical angles, θc,

for X-rays provide

atomic spacing, d.

Adapted from Fig.

3.2W, Callister 6e .

X-RAYS TO CONFIRM CRYSTAL STRUCTURE

reflections must be in phase todetect signal

spacingbetweenplanes

d

i n c o m i n g

X - r a y s

o u t g

o i n g

X - r a y

s

d e t e c t o r

θλ

θ

extradistancetravelledby wave “2”

“ 1 ” “ 2 ”

“ 1 ”

“ 2 ”

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Chapter 03 Annot