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Lecture of January 21

ENGR 4680U

Nuclear Materials

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Topics I

1. Structure of Crystalline Solids

2. Solidification and Defects

3. Alloys and Phase Diagrams

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Atoms in the Crystals

Want to give the position of atoms

Want to describe directions Want to describe planes of atoms

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Atom Position

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Bracket Convention

specific Family or

of a classdirection [ ] < >

plane ( ) { }

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Specific vs Family~Directions~

family of directions is

the following 8 specific directions:

[111], [111], [111], [111], [111], [111], [111], [111]

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Specific vs Family~Planes~

{100} family of planes is

the following 6 specific planes:

(100), (010), (001), (100), (010), (001),

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Indices of planes are h,k and l, the inverse of the intercepts on the a,b,and c axes respectively (0 for axis parallel to the plane, 1/)

Miller indices - planes

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Body-centred Cubic

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Body-centred Cubic

Inverse

of

intercepts

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Face-centred Cubic

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]2011[

]0121[

]3121[]0001[

]0011[

cph Crystal Structure

Note that for

cph structures

there is also a4 coordinate

system as well

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Angle Between Directions

given two directions [uvw] and [uvw]

angle between them is

let D be the vector of [uvw] and D is the vector of [uvw]

so:

cosD D D D

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Angle Between Directions

or

Note: this only applies to a cubic system

See handout for other systems

cos

cos

D D D D

D D

D D

2 2 2 2 2 2cos

uu vv ww

u v w u v w

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Braggs Law

2dhklsin= nl

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Braggs Law

2dhklsin= nl

Incident Beam

Reflected Beam

2d

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Neutron vs X-ray Diffraction

Neutrons interact with nucleus while X-rays

interact with electron cloud

Neutrons penetrating; X-rays absorbed near

surface

Neutrons can probe bulk properties

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Polycrystalline Zr

Magnification is: 350X

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Solid Solutions

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Solid Solutions

Hume-Rothery rules for solid solution formation:

less than about 15% difference in atomic radii

the same crystal structure

similar electronegativity values

same valence

all 4 rules must be satisfied; if any rule violated only

partial solubility possible

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Solid Solutions

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Substitutional Solid Solution

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Solid Solutions

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Interstitial Solid Solution

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Interstitial Solid Solution

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Point Defects

1. Vacancy

2. Self interstitial

3. Interstitial impurity

4. Substitutional impurity

under tension

5. Substitutional impurity

under compression

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Two Types of Point Defects

Schottky (German) defect involves a pair of

oppositely charged ion vacancies

Frenkel (Russian) defect is a vacancy-

interstitialcy combination

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self-interstitialatom (SIA)

Point Defects in Crystals

vacancy solute atoms

substitutional interstitial

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Thermal Production of Defects

for many processes, the process rate rises

exponentially with temperature

diffusivity of elements in metal alloys

rate of creep deformation

electrical conductivity

Diff i i S lid

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Diffusion in Solids

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General Equation

Arrhenius (Swedish) Equation

where:

C is a preexponential constant; Q is the activation energy;

R is the universal gas constant; T is the absolute temperature

k is Boltzmanns constant; q = Q/NA

kT

q

RT

Q

Cerate

T1

RQCln)rateln(

Cerate

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General Equation

I i i l Diff i

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Interstitial Diffusion


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jump frequency

flux to left/right

net flux

jump distance

diffusion coefficientunits of m2s-1

concentration gradient

xC

DJ

xC

61JJJ

n

6

1J

n6

1J

iii

i2

iiii

2ii

1ii

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Interstitial Jump Frequency

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InterstitialDiffusion

entropy/enthalpy of migrationvibration frequency

number of available sites

jump distance

kT

HDD

kT

H

k

Sz

mi

iiii

mimi

iii

exp

6

1

expexp

0

2

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Slip Lines

Side View Front View

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F di d h l l d

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Forces to cause coordinated shear were calculated

G

shear modulus

3 orders of

magnitude too large

shear strain

62

22sin

Ga

bG

G

b

x

b

x

m

mm

Sli S

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Slip Systems

At low temperatures / high stresses: Deformation of crystals

occurs exclusively by slip of lattice planes

Slip system is characterized by:

slip plane normal

slip direction

slip vector (a lattice vector in the slip direction)

often: slip planes are most densely packed lattice planes

slip directions are most densely packed lattice directions

n

s

Sli S

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Slip Systems

Sli S

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Slip Systems

Shear in close packed direction

easy

Shear in nonclose packed direction

harder

Sli S

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Slip Systems

slip frequency occurs on the most closelypacked planes, and almost invariably in the

most closely packed direction:

fcc ,{111}

bcc ,{110}

hcp ,{0002} and also {1010}

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Driving Force for Slip

Tensile stress, s= F/A

Resolved shear stress Rin

slip system

R= scos lcos fnsl

f

F

F A

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R= 0 if l= 90oor f= 90o

n

sl

F

F A

normal perpendicular

f

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R= max when l= f= 45o

n

sl

F

F A

f

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How does slip occur?

Note: slip is not a simultaneous breaking of

all bonds

Involves the motion of defects- dislocations

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Dislocations

Edge dislocation

Screw dislocation

The presence of defects (dislocations,

vacancies, etc.) explains the discrepancy

between the computed (i.e., theoretical)and real yield stress.

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Edge Dislocation

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Edge Dislocation

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Edge Dislocation

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Movement of an ED

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Movement of an ED

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Movement of an ED

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Movement of an ED

Side view of edge dislocation

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Dislocation Climb

An example of positive climb

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Burgers Vector

Characterizes a dislocation

Results from a mismatch in the closed

circuit around a dislocation

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Burgers Circuits

Edge Dislocation Screw Dislocation

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Burgers Vector

disloc

Burgers vector

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Dislocations

Edge dislocation

line is perpendicular to Burgers vector

Positive moves right; negative moves left

Screw dislocation:

line is parallel to Burgers vector

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Screw Dislocation

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Screw Dislocation

Left Hand Screw Dislocation

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Screw Dislocation

Right Hand Screw Dislocation

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References

R.E. Reed-Hill, Physical Metallurgy Principles, 2nd

Edition, PWS-Kent Publishing Co., (1973), Chapter 4.

D.A. Porter and K.E. Easterling, Phase Transformationsin Metals and Alloys, 2nd Edition, Stanley Thornes

(Publishers) Ltd., Chapter 3

J.F. Shackelford, Introduction to Materials Science for

Engineers, 3rdEdition, Macmillan PublishingCo.,(1992).

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Gibbs Phase Rule

P + F = C + 2

P represents # of phases

F represents # of degrees of freedom

C represents # of components

2 (limiting state variables: 1 for temperature, 1

for pressure)

Police Force = Cops + 2 Vehicles

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Definitions

Phasehomogeneous region

chemically and structurally homogeneous

Componentdistinct chemical substance fromwhich phase is formed

Degrees of Freedomnumber of independent

variables available to the system

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Reduction of State Variables

If pressure is fixed P + F = C + 1

If temperature is fixed P + F = C + 1

If both fixed P + F = C

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Unary Phase Diagrams

Iron

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Binary Phase Diagrams

Partial Binary Diagram for Pd-Rh

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y g

XRh

0.0 0.2 0.4 0.6 0.8 1.0

Pd Rh

1 atm

T

emperature(K)

1500

2000

2500

1500

2000

2500

Liquid

Solid solution (fcc)

2236

1827

2200

Liquid + fcc

SolidusLiquidus

C Ni Bi

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Cu-Ni Binary

Rh-Pd

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Rh Pd

Rh PdXPd

0.0 0.2 0.6 0.8 1.0

500

1000

1500

2000

2500

3000

500

1000

1500

2000

2500

3000

T

emperature(K) 2236.4

1827.1

Rh (fcc) + Pd (fcc)

0.4

1182.8

0.45

Liquid

Solid (fcc)

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T i l S lid S l ti

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Terminal Solid Solutions

T i l Mi l E l i

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Typical Microstructural Evolution

Binary Eutectic Phase Diagram

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Binary Eutectic Phase Diagram

T i l Mi t t l E l ti

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Peritectic Phase Diagram for Pd-Ru

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g

Liquid

Pd (fcc)

Pd (fcc) + Ru (cph)

Ru(cph)

e

e

Liquid + e

Liquid +

0.0 0.2 0.4 0.6 0.8 1.0

500

1000

1500

2000

2500

3000

500

1000

1500

2000

2500

3000

2607.0

1827.1

1866.4

Pd RuXRu

T

emperatu

re(K)

0.9230.185

0.10

Eutectic and Eutectoid

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Fe-C

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T pical Microstr ct ral E ol tion

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lecture 2 2013 jan 21

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