1. Structure of Metals

7/28/2019 1. Structure of Metals

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Lattice

Chapter I The Structure of Metals

A latticeis an infinite array of evenly spaced pointswhich are all similarly situated. Each points are

regarded as similarly situated in the rest of the lattice

appears the same, and in the same orientation when

viewed form them. (J.F. Nye, Physical Properties of Crystals)

3 D


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Chapter I The Structure of Metals-- Ideal Crystal

An ideal crystal is defined to be a body in which theatoms are arranged in a lattice. That is:

(a) the atomic arrangement appears the same, and in thesame orientation, when viewed from all the lattice points

(b). Lattice + Basis = crystal structure(c) the form and orientation of the lattice in an idealcrystal is independent of which point in the crystal ischosen as origin.

(d) an ideal crystal is infinite in extent; real crystals arenot only bounded, but also depart from the ideal crystalsby possessing occasional imperfections

(e) Continuity


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Unit cell choices

Primitive unit cell:

A parallelepiped having lattice points at its corners only

The primitive unit cell is not unique (such A, B, C.

In 3-D, like simple cubic structure)

Multiply primitive unit cell

A unit cell which has lattice points at the

centers of its faces, or at its body center,or occasionally at other positions, in

addition to the points at its corners (like

FCC, BCC, E, D, F etc.)

Chapter I The Structure of Metals-- Unit Cell

A unit cell containing only one lattice point


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Reflection Symmetry

In crystal,

1. The symmetrical arrangement of atoms can be described

formally in terms of elements of symmetry.

2. A symmetry operation moves or transforms an object in

such a way that after transformation it coincides with itself.

Mirror line (image

line) (2D)

Mirror plane

(image plane) (3D)


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J.F. Nye, Physical Properties of Crystals

Continuity Ideal crystal

Discontinuity Real crystal

lattice translation vector:

a, b, and c

lattice points translation :

=u1a+u2b+u3c

u: arbitrary integer

R= R+ for a ideal

lattice frame (ideal crystal )

Translational symmetry


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Rotational symmetry:


2- fold

6- fold4- fold

3- fold

5-fold ??


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??


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in a cubic unit cell

3 tetrad rotation axes

4 triad rotation axes

6 diad rotation axes


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Cubic

Tetragonal

Orthorhombic

Monoclinic

Triclinic

Hexagonal

Trigonal


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Unit cell and Body-centered cubic

www.parc.xerox.com/.../ comparison_of_cubic_packing.htm

BCC unit cell

http://members.tripod.com/~EppE/jpgs/bodcubic.jpgs

lattice

Two atoms per unit cell

Coordination number 8 (# of adjacent atoms)


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Unit cell and Body-centered cubic

The unit cell of the bcc structure consists of acube with an atom at each corner and an atom in

the center of the cube

Typical metals with a bcc unit cell are

Molybdenum, Tungsten and iron ( i.e. iron atroom temperature)

Close-pack direction: diagonal direction passingthrough the centered atom, cornered atom and

the atom at opposite corner


(please find two more.)


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Octahedral interstices in BCC: Octahedral interstices are

bounded by 6 atoms whose centers join up to make anoctahedron (a 8-sided figures); you can find 18 oct. inter.

sites in a BCC unit cell (6 at face center, the other 12 locate

at edges.)



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tetrahedral interstices in BCC: tetrahedral interstices are

bounded by 4 atoms whose centers join up to make antetrahedron (a 4-sided figures); you can find 24 tera. inter.

sites in a BCC unit cell (there are four tetrahedral sites on

each of 6 BCC faces)

MATTER-Introduction to point defects


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Difference in size: Tetrahedral interstice > Octahedral interstice

MATTER-Introduction to point defects


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Unit cell and Face-centered cubic

www.parc.xerox.com/.../ comparison_of_cubic_packing.htm

Four atoms per unit cell

Coordination number 12 (# of adjacent atoms)


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Unit cell and Face-centered cubic

FCC unit cell consists of a cubic structure

with an atom at every corner of the cube and an

atom at the center of each of the six faces.

Typical examples of metals with an F.C.C unitcell include Aluminum, Silver, Gold, Nickel and

iron (i.e. iron at high temperatures). (please findtwo more)

Features of F.C.C metals are ductile and good

electrical conductors.


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Octahedral interstices in FCC: Octahedral interstices are

bounded by 6 atoms whose centers join up to make an

octahedron (a 8-sided figures); you can find 13 oct. inter.

sites in a FCC unit cell (one at center of the unit cell, the

other 12 locate at edges.)

MATTER-Introduction to point defectsMATTER-Introduction to point defects


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MATTER-Introduction to point defects MATTER-Introduction to point defects


Tetrahedral interstices in FCC: tetrahedral interstices are

bounded by 4 atoms whose centers join up to make antetrahedron (a 4-sided figures); you can find 8 tetra. inter.

sites in a FCC unit cell Diagonal passes throughthe center of the

tetrahedral site


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Difference in size:Octahedral interstice > Tetrahedral interstice

(BCC: Tetra. size > Octa. Size)

FCC


BCC


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Home work:

Draw an tetrahedron in a FCC unit cell and

index the four planes on the tetrahedron (problem

1.6 in Reed-Hill)

Calculate the packing density (efficiency) of

FCC , BCC and SC

Packing efficiency= Voccupied / Vunit cell


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The close-packed planes in FCC are the {111} set, and the

close-packed directions are the [110] set.

By moving each atom off the corner

of a FCC unit cell on e.g., [001] plane,you can see four independent slip planes

in the unit cell respectively.Arizona state university


Close-pack plane and close-pack direction

Unit cell and Face Centered Cubic


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Chapter I The Structure of MetalsChapter I The Structure of Metals

Unit cell and Face Centered Cubic

{200}{111}

Close-pack planeLess close-pack plane

Close-pack plane


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This arrangement provides a close packing

system of atoms as illustrated.

6 atoms per unit cell, coordination No. is

12

Zinc, Magnesium, etc four-axis coordinate system


Unit cell and Close-packed hexagonal (HCP, orC.P.H.)


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Magnesium (Discovered 1808)

Atomic number12

Atomic weight24.305

Periodic table group An alkaline-earth metal

A simple substance is light metals of

silver white

Hexagonal closed pack structure

lattice constant a=0.32094nm

c=0.52103nm

c/a=1.6235

Melting point650

Boiling point1107

Specific gravity1.741

Young's modulus42000MPa

Stiffness16000MPa (Poissons ratio0.38)

line coefficient of expansion

26.9410-6/(20200)

Heat conduction rate

1.55J/cmsec(Al2.88J/cmsec)



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Digital-camera EXILIM EX-S3 made by CASIO

Adopted example,

magnesium alloy


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Magnesium alloy- Notebook computer


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4-digit Miller indices for C.P.H. planes and directions

three axes (a1, a2, a3) along

close-packed direction on

basal plane

the fourth axis is normal to

basal plane, called C axis

the unit of measurement

along a axis is a, along c axis

is c


a1

-a1

-a3

a3

a2-a2

-C1

C1

a

c


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Axial Ratios in Close-packed-hexagonal Metals

1.601.8861.8561.6241.568c/a

TiCdZnMgBeMetal


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a1

-a1

-a3

a3

a2-a2

-C1

C1Basal plane

(u v w t)=(a1 a2 a3 c)

(a/, a/, a/, c/1c)

unit of measurement for each axis

the intersection

(0001)?



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a1

-a1

-a3

a3

a2-a2

-C1

C1

Prism planes of type I

A

C

D

B

E

F

G

H

Plane ABCD


(u v w t)=(a1 a2 a3 c)

(a/-a, a/a, a/ , c/ )

the intersection

(1100)EFGH?

Prism planes of type I = {1100}


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Prism planes of type II

a1

-a1

-a3

a3

a2-a2

-C1

C1

A

D

C

B

E

F


Plane ABCD


(u v w t)=(a1 a2 a3 c)

(a/a, a/a, a/-0.5a, c/ )

the intersection

(1120)Prism planes of type II = {1120}


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a1

-a1

-a3

a3

a2-a2

-C1

C1

Pyramidal planes

A B

C

D

E

Type I, order II (ABC): (1012)

Type I, order I (ABD): (1011)

Type II, order I (AED) :(1121)

Type II, order II (AEC): (1122)

(u, v, w, t)=(a1, a2, a3, t)

a1+a2=-a3


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Four-digit Miller indices ofdirections for CPH

C=[0001], plane normal of basal plane

Looking on basal plane

[u v w t]=[a1 a2 a3 c],

u+v= -w still holds for determining direction

Diagonal axes, type I (e.g., a1 ?)

Any component along a1?


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Four-digit Miller indices ofdirections for CPH

Diagonal axes, type II (e.g., A1 ?)

A1

Diagonal axes, type II (e.g., A1)

perpendicular to type I

a2

Any component along a2?


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Three-digit Miller indices ofdirections and planes for CPH


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Interstitial sites--size and shape

BCCFCC

Atoms just fit in the interstitial site, the unit cell

not being distorted or expended

In real cases, interstitial atoms (C, N, O...) largerthan the site, leading to symmetric or nonsymmetric

expansion in the lattice


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R: radius of atom

r: radius of maximum interstice size

J.D. Verhoeven, Found. of Phys. Metall. p. 10

the maximum sized sphere that could be placed in the voids


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www.caton.org/images/ chem/TableP.gif



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Lattice being distorted because of the interstitial atoms

symmetrical or nonsymmetrical expansion ?

excess strain due to squeezed-in interstitial atom


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a2

2

a2

2

regular polyhedron

For close-packed crystal structures


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For close-packed crystal structures (regular polyhedron)

some of the octahedral voids in FCC

some of the tetrahedral voids in FCC

some of the tetrahedral voids

in CPH


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Interstitial sites in BCC crystal structurea

Octa. tetra.

a23

a


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R: radius of atom

r: radius of maximum interstice size

J.D. Verhoeven, Found. of Phys. Metall. p. 10



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. AD = 2(R+r) and AB =

2R=ADcos45.

2R=2Rcos45+2rcos45 and

r/R=(1-cos45)/cos45=0.414

FCC


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Home work

According to the above sketches in the relation between unit cell

length (a) and atomic radius(R), please show that: (1) the tetrahedral

site in BCC is 0.291 R; (2) the octahedral site in FCC is 0.414R.


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not close-packed

close-packed

2-D packing

e.g., (111) of FCC

(0001) basal planes of CPH


60o

Closed-Packing sequence


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Chapter I The Structure of Metals Closed-Packing sequence

A

AA

A

A

A

A B

BB

B

B

B

B

A

AA

A

A

A

A

C

CC

C

C

C

C

Two alternativestackings on top (or

bottom) of A


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C.R. Brooks, ASM (1982)

3-D packingB sites

C sites



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C.R. Brooks, ASM (1982)


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FCC : ABCABC.. CPH : ABAB..

Prof. L.H. Chen, class note, p. 71, NCKU



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Summary:

The FCC{111} planes look like the CPH {0001}.

The FCC system starts the same as the HCP but the third layer, ratherthan lining up with the atoms of the first layer, instead lines up with theother holes in the first layer. The fourth layer then lines up with the

atoms of the first layer, with sequence ABCABCABC.

The HCP system is formed by taking the given hexagonal array and

stacking the next layer so that it fits in one of the holes of the first layer.

The third layer then is placed so that its atoms line up with those of the

first layer, with the sequence ABABAB being repeated.


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http://www.techfak.uni-kiel.de/matwis/amat/elmat_en/kap_5/backbone/r5_1_2.html


Polycrystal vs. Single crystal


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A. Chemical discontinuity--leading to internalboundaries separating phases of different composition (or

different composition and structure)

B. Structural (crystal orientational) discontinuity--leading to internal boundaries between crystals of the

same phase, resulting from orientation difference

Polycrystal--internal boundaries within crystal

Polycrystal vs. Single crystal

General requirements

Meet eitherA orB Polycrystal

Ch I Th S f M l


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A. Chemical discontinuity


Al

Mg

Zn

Fe

Mn

Ch I Th S f M l


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B. Structural (crystal orientational) discontinuity

low angel boundary

tilt bound. twist bound. high angle boundary

Swalin, Thermo. Of Solids, Fig. 10.14, 15, 17

Ch t I Th St t f M t l


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grain size:27.6m grain size:12.2m

grain size:3.6m

Grain, grain size and

grain boundary

(magnesium alloy)


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Ch t I Th St t f M t l


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Basal Plane

(0002)

RD

ND

TD

Crystal arrangement of mass production materials

Anisotropy


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Anisotropic microstructures (e.g.. 5083 aluminum alloy)

100m

100m

100m

Anisotropy

Ch t I Th St t f M t l Th S hi P j i


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1 Gnomonic

2 Stereographic

3 Orthographic

Verhoeven, Fig. 1.15

the relationship between planes and directions

angles between the poles on the projections are

always the true angles between the normals of the

planes

To determine angles between planes and directions

The Stereographic Projection

Points A, B projected to different locations,

depending on projection method

----only Stereo. Projection meet the requirement

Chapter I The Str ct re of Metals


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Imaging the plane passing thru the

center of a hemisphere

Project the intersection of the plane

and hemisphere onto project plane

http://www.telusplanet.net/public/nstuart/proj.gif

http://www.stmarys.ca/academic/science/geology/structural/stereoalt.htm

2-D drawing of 3-D crystallographic features

A lattice plane (3D) can be represented in

projection paper (2-D) by the normal of the lattice

plane and (or) the trace of the plane on ref. sphere

Chapter I The Structure of Metals Th St hi P j tiChapter I The Structure of Metals


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Chapter I The Structure of Metals The Stereographic ProjectionChapter I The Structure of Metals

Plane(passing through

the center of the sphere)

plane

normalIntersection with ref. hemisphere

Project the Poles and traces to project plane (equatorial plane)

Chapter I The Structure of MetalsChapter I The Structure of Metals


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: plane normal of (h k l)

(hkl)

n

P P: pole of (h k l)Trace of (h k l);

a great circle

Chapter I The Structure of MetalsChapter I The Structure of Metals The Stereographic Projection

Great circle : a circle of maximum diameter, if the plane passes

through the center of the sphere

Small circle : A plane not passes through the center of the

sphere will intersect the sphere in a small circleOn a ruled globe (see next page), the longitude lines

(meridians)--Great circles; the latitude lines (except equator) --

small circles



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2o intervals

Wulff Net (stereographic projection)



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Kelly and Groves, Fig. 2.2

Gnomonic

Stereographic

Orthographic

(h k l) plane

Pole of (h k l)



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Stereographic projection

Fig. 2-28, Cullity


Chapter I The Structure of Metals Stereographic Projection


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Stereographic Projection



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Reed-Hill, Fig. 1.22, p. 20.



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directions that lie in a plane




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Planes of a zone

Those planes that mutually intersect along a

common direction forms the planes of a zone.

The line of the intersection by those plane ofthe same zone is called the zone axis.

The direction of the zone axis is perpendicular

to each normal of the planes in the same zone.



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Planes of a zone


Reed-Hill, Fig. 1.25

zone axis [111]


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zone axis [111]




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A plane can be represented by its pole.

The direction of the zone axis is perpendicular

to each normal of the planes in the same zone.



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C apte e St uctu e o eta s



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Angle between planes and Wulff Net--How to determine

the angle between planes or directions by means of a 2-D

stereographic projection

p



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A

A

B

C

B

C



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p

All meridians (longitude

lines), including the basic

circle, are great circles.

The equator is a great

circle. All other latitude

lines are small circles.

Angular between points

representing directions in

space can be measured on

the Wulff net only in the

points are made to

coincide with a great

circle of the net.



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Angle between 2 planes



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Chapter I The Structure of Metals --Traces of P1 and P2


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Chapter I The Structure of Metals Traces of P1 and P2

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