The Muppet’s Guide to:The Structure and Dynamics of Solids

Phase Diagrams

Phase Diagrams• Indicate phases as function of T, Co, and P. • For this course: -binary systems: just 2 components.

-independent variables: T and Co (P = 1 atm is almost always used).

• PhaseDiagramfor Cu-Niat P=1 atm.

• 2 phases: L (liquid)

a (FCC solid solution)

• 3 phase fields: LL + aa

wt% Ni20 40 60 80 10001000

1100

1200

1300

1400

1500

1600T(°C)

L (liquid)

a (FCC solid solution)

L + aliquidus

solid

us

Figure adapted from Callister, Materials science and engineering, 7 th Ed.

Phase Diagrams• Indicate phases as function of T, Co, and P. • For this course: -binary systems: just 2 components.

-independent variables: T and Co (P = 1 atm is almost always used).

• PhaseDiagramfor Cu-Niat P=1 atm.

wt% Ni20 40 60 80 10001000

1100

1200

1300

1400

1500

1600T(°C)

L (liquid)

a (FCC solid solution)

L + aliquidus

solid

us

Figure adapted from Callister, Materials science and engineering, 7 th Ed.

Liquidus:Separates the liquid from the mixed L+ aphase

Solidus:Separates the mixed L+ a phase from the solid solution

wt% Ni20 40 60 80 10001000

1100

1200

1300

1400

1500

1600T(°C)

L (liquid)

a (FCC solid solution)

L + a

liquidus

solid

us

Cu-Niphase

diagram

Number and types of phases• Rule 1: If we know T and Co, then we know: - the number and types of phases present.

• Examples:

A(1100°C, 60): 1 phase: a

B(1250°C, 35): 2 phases: L + a

B (

1250

°C,3

5) A(1100°C,60)

Figure adapted from Callister, Materials science and engineering, 7 th Ed.

wt% Ni20

1200

1300

T(°C)

L (liquid)

a(solid)L + a

liquidus

solidus

30 40 50

L + a

Cu-Ni system

Composition of phases• Rule 2: If we know T and Co, then we know: --the composition of each phase.

• Examples:TA

A

35Co

32CL

At TA = 1320°C:

Only Liquid (L) CL = Co ( = 35 wt% Ni)

At TB = 1250°C:

Both a and L CL = C liquidus ( = 32 wt% Ni here)

Ca = C solidus ( = 43 wt% Ni here)

At TD = 1190°C:

Only Solid ( a) Ca = Co ( = 35 wt% Ni)

Co = 35 wt% Ni

BTB

DTD

tie line

4Ca3

Figure adapted from Callister, Materials science and engineering, 7 th Ed.

wt% Ni20

1200

1300

30 40 50110 0

L (liquid)

a (solid)

L + a

L + a

T(°C)

A

35Co

L: 35wt%Ni

Cu-Nisystem

• Phase diagram: Cu-Ni system.

• System is: --binary i.e., 2 components: Cu and Ni. --isomorphous i.e., complete solubility of one component in another; a phase field extends from 0 to 100 wt% Ni.

• Consider Co = 35 wt%Ni.

Cooling a Cu-Ni Binary - Composition

4635

4332

a: 43 wt% Ni

L: 32 wt% Ni

L: 24 wt% Ni

a: 36 wt% Ni

Ba: 46 wt% NiL: 35 wt% Ni

C

D

E

24 36

Figure adapted from Callister, Materials science and engineering, 7 th Ed. USE LEVER RULE

• Tie line – connects the phases in equilibrium with each other - essentially an isotherm

The Lever Rule – Weight %

How much of each phase? Think of it as a lever

ML M

R S

RMSM L

L

L

LL

LL CC

CC

SR

RW

CC

CC

SR

S

MM

MW

00

wt% Ni

20

1200

1300

T(°C)

L (liquid)

a(solid)L + a

liquidus

solidus

30 40 50

L + aB

T B

tie line

CoC L Ca

SR

Figure adapted from Callister, Materials science and engineering, 7 th Ed.

• Rule 3: If we know T and Co, then we know: --the amount of each phase (given in wt%).• Examples:

At T A : Only Liquid (L) W L = 100 wt%, W a = 0

At T D : Only Solid ( a) W L = 0, W a = 100 wt%

C o = 35 wt% Ni

Weight fractions of phases – ‘lever rule’

wt% Ni20

1200

1300

T(°C)

L (liquid)

a(solid)L + a

liquidus

solidus

30 40 50

L + a

Cu-Ni system

TA A

35C o

32C L

BT B

DT D

tie line

4Ca3

R S

= 27 wt%

43 3573 %

43 32wt

At T B : Both a and L

WL= S

R + S

Wa= R

R + S

Figure adapted from Callister, Materials science and engineering, 7 th Ed.

wt% Ni20

120 0

130 0

30 40 50110 0

L (liquid)

a (solid)

L + a

L + a

T(°C)

A

35C o

L: 35wt%Ni

Cu-Nisystem

• Phase diagram: Cu-Ni system.• System is: --binary i.e., 2 components: Cu and Ni. --isomorphous i.e., complete solubility of one component in another; a phase field extends from 0 to 100 wt% Ni.• Consider Co = 35 wt%Ni.

Cooling a Cu-Ni Binary – wt. %

46344332

a: 27 wt%

L: 73 wt%

L: 8 wt%

a: 92 wt%

Ba: 8 wt% L: 92 wt%

C

D

E

24 36

Figure adapted from Callister, Materials science and engineering, 7 th Ed.

Equilibrium cooling

• Multiple freezing sites– Polycrystalline materials– Not the same as a single crystal

• The compositions that freeze are a function of the temperature

• At equilibrium, the ‘first to freeze’ composition must adjust on further cooling by solid state diffusion

Diffusion is not a flow

Our models of diffusion are based on a random walk approach and not a net flow

http://mathworld.wolfram.com/images/eps-gif/RandomWalk2D_1200.gif

Concept behind mean free path in scattering phenomena - conductivity

Diffusion in 1 Dimension

• Fick’s First Law

dCJ D T

dx

J = flux – amount of material per unit area per unit timeC = concentration

Diffusion coefficient which we expect is a function of the temperature, T

Diffusion cont….• Requires the solution of the continuity equation:

The change in concentration as a function of time in a volume is balanced by how much material flows in per time unit minus how much flows out – the change in flux, J:

• giving Fick’s second law (with D being constant):

2

2

C C CD D T

t x x x

0C Jt x

dC

J D Tdx

BUT

Solution of Ficks’ Laws

C

x

CCo

t = 0

t = t

For a semi-infinite sample the solution to Ficks’ Law gives an error function distribution whose width increases with time

Consider slabs of Cu and Ni.

Interface region will be a mixed alloy (solid solution)

Interface region will grow as a function of time

wt% Ni20

120 0

130 0

30 40 50110 0

L (liquid)

a (solid)

L + a

L + a

T(°C)

A

35C o

L: 35wt%Ni

Cu-Nisystem

Co = 35 wt%Ni.

Slow Cooling in a Cu-Ni Binary

a: 43 wt% Ni

L: 32 wt% Ni

L: 24 wt% Ni

a: 36 wt% Ni

Ba: 46 wt% NiL: 35 wt% Ni

C

D

E

Figure adapted from Callister, Materials science and engineering, 7 th Ed.

Enough time is allowed at each temperature change for atomic diffusion to occur. – Thermodynamic ground state

Each phase is homogeneous

Non – equilibrium

cooling α

L

α + L

Figure adapted from Callister, Materials science and engineering, 7 th Ed.

Reduces the melting

temperature

No-longer in the thermodynamic

ground state

• Ca changes as we solidify.• Cu-Ni case:

• Fast rate of cooling: Cored structure

• Slow rate of cooling: Equilibrium structure

First a to solidify has Ca = 46 wt% Ni.

Last a to solidify has Ca = 35 wt% Ni.

Cored vs Equilibrium Phases

First a to solidify: 46 wt% Ni

Uniform C a:

35 wt% Ni

Last a to solidify: < 35 wt% Ni

Figure adapted from Callister, Materials science and engineering, 7 th Ed.

2 componentshas a special compositionwith a min. melting temperature

Binary-Eutectic Systems – Cu/Ag

• 3 phases regions, L, a and b and 6 phase fields - L, a and , b L+ , a L+ , +b a b

• Limited solubility – mixed phases

Figure adapted from Callister, Materials science and engineering, 7 th Ed.

a phase:

Mostly copper

b phase:

Mostly Silver

Solvus line – the solubility limit

Min. melting TE

Binary-Eutectic Systems

• Eutectic transitionL(CE) (CE) + (CE)

• TE : No liquid below TE

TE, Eutectic temperature, 779°CCE, eutectic composition, 71.9wt.%

The Eutectic point

Cu-Ag system

L (liquid)

a L + a L +b b

a + b

Co wt% Ag in Cu/Ag alloy20 40 60 80 1000

200

1200T(°C)

400

600

800

1000

CE

TE CaE=8.0 CE=71.9 CbE=91.2

779°C

Figure adapted from Callister, Materials science and engineering, 7 th Ed.

Any other composition, Liquid transforms to a mixed L+solid phase

E

L + aL +b

a + b

200

T(°C)

18.3

C, wt% Sn20 60 80 1000

300

100

L (liquid)

a 183°C 61.9 97.8

b

• For a 40 wt% Sn-60 wt% Pb alloy at 150°C, find... --the phases present: Pb-Sn

system

Pb-Sn (Solder) Eutectic System (1)

a + b--compositions of phases:

CO = 40 wt% Sn

--the relative amount of each phase:

150

40

Co

11

C

99

C

SR

Ca = 11 wt% SnCb = 99 wt% Sn

Wa =C - CO

C - C

= 99 - 4099 - 11

= 5988 = 67 wt%

SR+S =

W =CO - C

C - C=R

R+S

= 2988

= 33 wt%= 40 - 1199 - 11

Figure adapted from Callister, Materials science and engineering, 7 th Ed.

• 2 wt% Sn < Co < 18.3 wt% Sn• Result: Initially liquid → liquid + then alone finally two phases

a poly-crystal fine -phase inclusions

Microstructures in Eutectic Systems: II

Pb-Snsystem

L + a

200

T(°C)

Co , wt% Sn10

18.3

200Co

300

100

L

a

30

a + b

400

(sol. limit at TE)

TE

2(sol. limit at T room)

La

L: Co wt% Sn

ab

a: Co wt% Sn

Figure adapted from Callister, Materials science and engineering, 7 th Ed.

• Result: Eutectic microstructure (lamellar structure) --alternating layers (lamellae) of a and b crystals.

Microstructures in Eutectic Systems: Co=CE

160 m

Micrograph of Pb-Sn eutectic microstructurePb-Sn

systemL

a

200

T(°C)

C, wt% Sn

20 60 80 1000

300

100

L

a b

L+ a

183°C

40

TE

18.3

: 18.3 wt%Sn

97.8

: 97.8 wt% Sn

CE61.9

L: Co wt% Sn

Figures adapted from Callister, Materials science and engineering, 7 th Ed.

• Result: Eutectic microstructure (lamellar structure) --alternating layers (lamellae) of a and b crystals.

Microstructures in Eutectic Systems: Co=CE

Pb-Snsystem

L

a

200

T(°C)

C, wt% Sn

20 60 80 1000

300

100

L

a b

L+ a

183°C

40

TE

18.3

: 18.3 wt%Sn

97.8

: 97.8 wt% Sn

CE61.9

L: Co wt% Sn

Figures adapted from Callister, Materials science and engineering, 7 th Ed.

97.8 61.945.2%

97.8 18.3W

61.9 18.354.8%

97.8 18.3W

Pb rich

Sn Rich

Lamellar Eutectic Structure

Figure adapted from Callister, Materials science and engineering, 7 th Ed.

PbSn

At interface, Pb moves to a-phase and Sn migrates to b- phase

Lamellar form to minimise diffusion distance – expect spatial extent to depend on D and cooling rates.

• 18.3 wt% Sn < Co < 61.9 wt% Sn• Result: a crystals and a eutectic microstructure

Microstructures IV

18.3 61.9

SR WL = (1- Wa) = 50 wt%

Ca = 18.3 wt% Sn

CL = 61.9 wt% SnS

R + SWa = = 50 wt%

• Just above TE :

Pb-Snsystem

L+b200

T(°C)

Co, wt% Sn

20 60 80 1000

300

100

L

a b

L +a

40

a +b

TE

L: Co wt% Sn LL

Figure adapted from Callister, Materials science and engineering, 7 th Ed.

• 18.3 wt% Sn < Co < 61.9 wt% Sn• Result: a crystals and a eutectic microstructure

Microstructures IV

18.3 61.9

SR

97.8

SR

Primary, a

Eutectic, a Eutectic, b

• Just below TE :Ca = 18.3 wt% SnCb = 97.8 wt% Sn

SR + S

Wa = = 73 wt%

Wb = 27 wt%

Pb-Snsystem

L+b200

T(°C)

Co, wt% Sn

20 60 80 1000

300

100

L

a b

L +a

40

a +b

TE

L: Co wt% Sn LL

Figure adapted from Callister, Materials science and engineering, 7 th Ed.

Intermetallic Compounds

Mg2Pb

Note: intermetallic compound forms a line - not an area - because stoichiometry (i.e. composition) is exact.

Figure adapted from Callister, Materials science and engineering, 7 th Ed.

a phase:

Mostly Mg

b phase:

Mostly Lead

Eutectoid & Peritectic

Cu-Zn Phase diagram

Eutectoid transition +

Peritectic transition + L

Figure adapted from Callister, Materials science and engineering, 7 th Ed.

mixed liquid and solid to single solid transition

Solid to solid ‘eutectic’ type transition

Iron-Carbon (Fe-C) Phase Diagram• 2 important points

-Eutectoid (B): g a + Fe3C

-Eutectic (A): L g + Fe3C

Fe3C

(cem

entit

e)

1600

1400

1200

1000

800

600

4000 1 2 3 4 5 6 6.7

L

g(austenite)

g+L

g +Fe3C

a +Fe3C

a+g

L+Fe3C

d

(Fe) Co, wt% C

1148°C

T(°C)

a 727°C = T eutectoid

ASR

4.30Result: Pearlite = alternating layers of

a and Fe3C phases

120 mm

g ggg

R S

0.76

Ceu

tect

oid

B

Fe3C (cementite-hard)a (ferrite-soft)

Figure adapted from Callister, Materials science and engineering, 7 th Ed.

Iron-Carbon

http://www.azom.com/work/pAkmxBcSVBfns037Q0LN_files/image003.gif

The Muppet’s Guide to:The Structure and Dynamics of Solids

The Final Countdown

CharacterisationOver the course so far we have seen how thermodynamics plays an important role in defining the basic minimum energy structure of a solid.

Small changes in the structure (such as the perovskites) can produce changes in the physical properties of materials

Kinetics and diffusion also play a role and give rise to different meta-stable structures of the same materials – allotropes / polymorphs

Alloys and mixtures undergo multiple phase changes as a function of temperature and composition

BUT how do we characterise samples?

Probes

Resolution better than the inter-atomic spacings

• Electromagnetic Radiation

• Neutrons

• Electrons

Probes

Treat all probes as if they were waves:

;

hp k p mv

Wave-number, k:2

k k

Momentum, p:

Photons ‘Massive’ objects

Xavier the X-ray

hcE

Ex(keV)=1.2398/l(nm)

Speed of Light

Planck’s constant Wavelength

Elastic scattering as Ex>>kBT

Norbert the Neutron

hmv

222

12 2n

n

hE mvm

En(meV)=0.8178/l2(nm)

De Broglie equation:

mass velocity

Kinetic Energy:

Strong inelastic scattering as En~kBT

Eric the Electron

• Eric’s rest mass: 9.11 × 10−31 kg.• electric charge: −1.602 × 10−19 C• No substructure – point particle

hmvDe Broglie equation:

mass velocity

Ee depends on accelerating voltage :– Range of Energies from 0 to MeV

ProbesResolution better than the interatomic spacings

Absorption low – we want a ‘bulk’ probe

• Electrons - Eric

• quite surface sensitive

• Electromagnetic Radiation - Xavier

• Optical – spectroscopy

• X-rays :

• VUV and soft (spectroscopic and surfaces)

• Hard (bulk like)

• Neutrons - Norbert

Interactions

1. Absorption

2. Refraction/Reflection

3. Scattering Diffraction

EnglebertXavier

Norbert

Crystals are 2D with planes separated by dhkl. There will only be constructive interference when == - i.e. the reflection

condition.

a

Basic Scattering Theory

The number of scattered particles per

second is defined using the standard

expression

I Id

ds 0

Unit solid angle Differentialcross-section

Defined using Fermi’s Golden Rule

INTERACTION POTENTId

dFi Ana L i ll Init a

Spherical Scattered Wavefield

ScatteringPotential

Incident Wavefield

Different for X-rays, Neutrons and Electrons

2

exp k r r r r

d

dd

i V

BORN approximation:• Assumes initial wave is also spherical• Scattering potential gives weak interactions

0

2rexp kp rk r rex V i

ddi

d

2

exp q rr r id

dd

V

Scattered intensity is proportional to the Fourier Transform of the scattering potential

q k k0

2

exp k r r r rd

i V dd

Scattering from CrystalAs a crystal is a periodic repetition of atoms in 3D we can formulate the scattering amplitude from a crystal by expanding the scattering

from a single atom in a Fourier series over the entire crystal

( ) exp q r V

f r i dV

(q) q exp q T rj jT j

A fi

Atomic Structure Factor

Real Lattice Vector: T=ha+kb+lc

The Structure FactorDescribes the Intensity of the diffracted beams in reciprocal space

exp q r exp u v w 2jj j

i i h k l

hkl are the diffraction planes, uvw are fractional co-ordinates within

the unit cell

If the basis is the same, and has a scattering factor, (f=1), the structure

factors for the hkl reflections can be found hkl

Weight phase

The Form Factor

Describes the distribution of the diffracted beams in reciprocal space

The summation is over the entire crystal which is a parallelepiped of sides:

1

1

32

2 3

1T 1

2 31 1

q exp q T exp q a

exp q b exp q c

N

n

NN

n n

L i n i

n i n i

1 2 3N a N b N c

The Form FactorMeasures the translational symmetry of the lattice

The Form Factor has low intensity unless q is a

reciprocal lattice vector associated with a reciprocal

lattice point

1,2,3 1,2,3 1,2,3

sin s sin sq exp s

sin s

i

i

Ni i i i

i ijini i i

N NL i n

s

0

0.5x105

1.0x105

1.5x105

2.0x105

2.5x105

-0.02 -0.01 0 0.01 0.02

Deviation parameter, s1 (radians)

[L(s

1)]

2

N=2,500; FWHM-1.3”

N=500

q d s Deviation from reciprocal lattice point located at d*

Redefine q:

The Form Factor

0

20

40

60

80

100

-0.6 -0.3 0 0.3 0.6

Deviation parameter, s1 (radians)

[L(s

1)]

2

0

0.5x105

1.0x105

1.5x105

2.0x105

2.5x105

-0.02 -0.01 0 0.01 0.02

Deviation parameter, s1 (radians)

[L(s

1)]

2

The square of the Form Factor in one dimension

N=10 N=500

1,2,3

sin sq i i

ji

NL

s

Scattering in Reciprocal Space

T

q q exp q r exp q Tj jj

A f i i Peak positions and intensity tell us about the structure:

POSITION OF PEAK

PERIODICITY WITHIN SAMPLE

WIDTH OF PEAK

EXTENT OF PERIODICITY

INTENSITY OF PEAK

POSITION OF ATOMS IN

BASIS

Powder DiffractionIt is impossible to grow some materials in a single crystal form or

we wish to study materials in a dynamic process.

Powder Techniques

Allows a wider range of materials to be studied under different sample conditions

1. Inductance Furnace 290 – 1500K

2. Closed Cycle Cryostat 10 – 290K

3. High Pressure Up-to 5 million Atmospheres

• Phase changes as a function of Temp and Pressure

• Phase identification

Search and MatchPowder Diffraction often used to identify phases

Cheap, rapid, non-destructive and only small quantity of sample

Inte

nsi

ty

2 A ngle

JCPDS Powder Diffraction File lists materials (>50,000) in order of their d-

spacings and 6 strongest reflectionsOK for mixtures of up-to 4

components and 1% accuracy

Monochromatic x-rays

Diffractometer

High Dynamic range detector

Single Crystal Diffraction

2dhkl hklsin Monochromatic radiation so sample needs to moved to the

Bragg condition….

Angular resolution is the Darwin width of analyser crystal (Typically 10-20”)

Detailed Lateral Information obtained

XMaS Beamline - ESRF

StrainPeak positions defined by the lattice parameters:

1

1 1, ,

q exp qN

ini a b c

L i n

Strain is an extension or compression of the lattice,

hkl hkld d

Results in a systematic shift of all the peaks

Ho Thin FilmsXRD measured as a function of temperature

10-4

10-2

100

102

104

20 40 60 80 100

T=294KT=244KT=194KT=94KT=144KT=42KT=10KT=300K

Scattering Angle ()

Inte

nsity

(ar

b. u

nits

)

Ho Thin FilmsSubstrate and Ho film follow have different behaviour

1

10

100

1000

30 32 34 36

T=294KT=244KT=194KT=144KT=94KT=42KT=10K

Scattering Angle ()

Inte

nsi

ty (

arb

. u

nits

)

Whole film refinement

Peak BroadeningDiffraction peaks can also be broadened in qz by:

1. Grain Size 2. Micro-Strains OR Both

The crystal is made up of particulates which all act as perfect but small crystals

, ,

sin sq i i

ii a b c

NL

s

Number of planes sampled is finite

Recall form factor: Scherrer Equation

2

cosSizeBD

NixMn3-xO4+ (400 Peak)As Grown at 200ºC AFM images (1200 x 1200 nm)

400

0

0.2

0.4

0.6

0.8

1.0

-1.0 -0.5 0 0.5 1.0

900C850C

800C

750C700C

650C

2

Inte

nsi

ty

D

450nm thick films

Annealed at 800ºC

Peak BroadeningDiffraction peaks can also be broadened in qz by:

1. Grain Size 2. Micro-Strains OR Both

The crystal has a distribution of inter-planar spacings dhkl ±Ddhkl.

Diffraction over a range, ,Dq of angles

Differentiate Bragg’s Law: 2 2 tanStrain B

Width in radians

Strain Bragg angle

dd

Peak BroadeningDiffraction peaks can also be broadened in qz by:

1. Grain Size 2. Micro-Strains OR Both

Total Broadening in 2q is sum of Strain and Size:

2 2 tancosTotal B

BD

2 cos 2 sinhkl hkl hklB B D

Rearrange

Williamson-Hall plot

y mx c

Powder Diffraction

0

100

200

300

400

30 40 50 60

Detector Angle (°)

Inte

nsity

(a

rb.

units

)

Powder of Nickel ManganiteCUBIC Structure

0

0.05

0.10

0.15

0.20

0.25

0 10 20 30

333422

400

222311

220

y=(1.5412/(4*a2))xa=8.348 ± 0.0036

(h2+k2+l2)

sin2

(B)

0.005

0.006

0.007

0.008

0.05 0.10 0.15 0.20 0.25

y=((1.541/d))+(2s)xGrain Size=299 ± 19.5a/a = 0.005 ± 0.001

sin(B)

Wid

th *

cos

(B) Grain size = 30±2nm

Strain Dispersion = 0.005±0.001

Cubic-Tetragonal Distortions

CUBICTETRAGONAL

a c a c

High Temperature Powder XRD

25 30 35 40 45 50 55 60 65 702Theta (°)

0

10000

20000

30000

Inte

ns

ity

(c

ou

nts

)

30.8 30.9 31.0 31.1 31.2 31.3 31.4 31.5 31.6 31.72Theta (°)

10000

20000

30000

Inte

ns

ity

(c

ou

nts

)

0.4BiSCO3 - 0.6PbTiO3 (K. Datta)

Tetragonal → Cubic phase transition

Courtesy, D. Walker and K. Datta University of Warwick

CsCoPO4

Dr. Mark T. Weller, Department of Chemistry, University of Southampton, www.rsc.org/ej/dt/2000/b003800h/

Variable temperature powder X-ray diffraction data show a marked change in the pattern at 170 °C.

Sn in a Silica Matrix

1. What form of tin

2. Particle size

3. Strain

4. Melting Temperature

Eutectic’s

wt% Ni20

120 0

130 0

30 40 50110 0

L (liquid)

a (solid)

L + a

L + a

T(°C)

A

35C o

L: 35wt%Ni

Cu-Nisystem

• Consider Cu/Ni with 35 wt.% Ni

Following Structural Changes

4332

a: 43 wt% Ni

L: 32 wt% Ni

L: 24 wt% Ni

a: 36 wt% Ni

Ba: 46 wt% NiL: 35 wt% Ni

C

D

E

24 36

Figure adapted from Callister, Materials science and engineering, 7 th Ed. USE LEVER RULE

A. Liquid

B. Mixed Phase

C.

D.

E. Solid

Cored Samples

α

L

α + L

Issues:

Lattice Parameter

Particle Size

Strain Dispersion

2 cos 2 sinhkl hkl hklB B D

NiCrStructural Changes?

Fcc: hkl are either all odd or all

even.

Bcc: sum of hkl must be even.