Crystallographic Planes Miller Indices: Reciprocals of the (three) axial intercepts for a plane,...

Crystallographic Planes• Miller Indices:

Reciprocals of the (three) axial intercepts for a plane, cleared of fractions & common multiples. All parallel planes have same Miller indices.

• Algorithm 1. Read off intercepts of plane with axes in terms of a, b, c2. Take reciprocals of intercepts3. Reduce to smallest integer values4. Enclose in parentheses, no commas i.e., (hkl)

Crystallographic Planesz

x

ya b

c

4. Miller Indices (110)

Example 2 a b cz

x

ya b

c

4. Miller Indices (200)

1. Intercepts 1 1 2. Reciprocals 1/1 1/1 1/

1 1 03. Reduction 1 1 0

1. Intercepts ½ 2. Reciprocals 1/½ 1/ 1/

2 0 03. Reduction 2 0 0

Example 1 a b c

½

Crystallographic Planesz

x

ya b

c

4. Miller Indices (634)

Example 31. Intercepts ½ 1 ¾

a b c

2. Reciprocals 1/½ 1/1 1/¾2 1 4/3

3. Reduction 6 3 4

• Single Crystals-Properties vary with direction: anisotropic.

-Example: the modulus of elasticity (E) in BCC iron:

Data from Table 3.3, Callister 7e.(Source of data is R.W. Hertzberg, Deformation and Fracture Mechanics of Engineering Materials, 3rd ed., John Wiley and Sons, 1989.)

Single Crystals

E (diagonal) = 273 GPa

E (edge) = 125 GPa

Fracture surface of failed stainless steel bolt

200X – As cast aluminum from exercise equipment

•Properties may/may not vary with direction.

•If grains are randomly oriented: isotropic

(Epoly iron = 210 GPa)

•If grains are textured: anisotropic200 m

Polycrystalline Materials

Crystal Defects

• Vacancies• Interstitial atoms• Substitutional atoms

Point defects

• Dislocations Linear defects

• Grain Boundaries Interfacial defects

• There is no such thing as a perfect crystal• We use/engineer the imperfections to control properties

Types of Defects

• Vacancies:-vacant atomic sites in a structure.

• Self-Interstitials:-"extra" atoms positioned between atomic sites.

Point Defects

Vacancydistortion of planes

self-interstitial

distortion of planes

Boltzmann's constant

(1.38 x 10 -23 J/atom-K)

(8.62 x 10-5 eV/atom-K)

Nv

Nexp

Qv

kT

No. of defects

No. of potential defect sites.

Activation energy

Temperature

Each lattice site is a potential vacancy site

• Equilibrium concentration varies with temperature!

Equilibrium Concentration:Point Defects

Notes:Form is of an Arrehnius EquationPV=nRT; n is the number of molesPV=NkT; N is the number of molecules

k = R/Na

• We can get Qv from an experiment.

Nv

N= exp

Qv

kT

Measuring Activation Energy

• Measure this...

Nv

N

T

exponential dependence!

defect concentration

• Replot it...

1/T

N

Nvln

-Qv /k

slope

• Find the equil. # of vacancies in 1 m3 of Cu at 1000C.• Given:

ACu = 63.5 g/mol = 8.4 g/cm3

Qv = 0.9 eV/atom NA = 6.02 x 1023 atoms/mol

Estimating Vacancy Concentration

For 1 m3 , N =NAACu

x x 1 m3 = 8.0 x 1028 sites8.62 x 10-5 eV/atom-K

0.9 eV/atom

1273K

Nv

Nexp

Qv

kT

= 2.7 x 10-4

• Answer:

Nv = (2.7 x 10-4)(8.0 x 1028) sites = 2.2 x 1025 vacancies

• Low energy electron microscope view of a (110) surface of NiAl.• Increasing T causes surface island of atoms to grow.• Why? The equil. vacancy conc. increases via atom motion from the crystal to the surface, where they join the island.

Reprinted with permission from Nature (K.F. McCarty, J.A. Nobel, and N.C. Bartelt, "Vacancies inSolids and the Stability of Surface Morphology",Nature, Vol. 412, pp. 622-625 (2001). Image is5.75 m by 5.75 m.) Copyright (2001) Macmillan Publishers, Ltd.

Observing Equilibrium Vacancy Conc.

Island grows/shrinks to maintain equil. vacancy conc. in the bulk.

Alloys

• Book makes the point that cannot truly refine to a purity level greater than 99.9999% (“4 9s”)– We just calculated that there are 8 x 1028 atoms in one m3 of Cu– Multiply 0.0001% by 8 x 1028 = 8 x 1022 Impurity Atoms

• In real world we don’t work with pure metals but rather we work with alloys

• Alloys are “metal soup” in which impurities have been added intentionally (or unintentionally through processing) to produce specific properties

Two outcomes if impurity (B) added to host (A):• Solid solution of B in A (i.e., random dist. of point defects)

• Solid solution of B in A plus particles of a new phase (usually for a larger amount of B)

OR

Substitutional solid soln.(e.g., Cu in Ni)

Interstitial solid soln.(e.g., C in Fe)

Second phase particle--different composition--often different structure.

Solid Solutions

BCC Interstitial Sites

Source: C. Barrett and T.B. Massalski, Structure of Metals, 3rd Revised Edition, Pergamon, 1980.

FCC Interstitial Sites

Source: C. Barrett and T.B. Massalski, Structure of Metals, 3rd Revised Edition, Pergamon, 1980.

Conditions for Substitutional Solid Solutions

Hume – Rothery Rules1. r (atomic radius) < 15%

2. Proximity in periodic table i.e., similar electronegativities. If the difference in electronegativity is too great will tend to form intermetallic compounds instead of solid solutions

3. Same crystal structure for pure metals

4. Similar ValencyMetals have a greater tendency to dissolve metals of higher valency than lower valency

William Hume-Rothery was a British Metallurgist who founded the Metallurgy Department at Oxford in the 1950s

Conditions for Interstitial Solid Solutions

Hume – Rothery Rules1. Solute atoms must be similar in size to the

interstitial locations in lattice structure

2. Proximity in periodic table i.e., similar electronegativities

Application of Hume–Rothery Rules

1. Would you predictmore Al or Ag to dissolve in Zn?

2. More Zn or Al

in Cu?

3. Will C form substitutional or interstitial solid solution with iron? Table on p. 106, Callister 7e.

Element Atomic Crystal Electro- ValenceRadius Structure nega-

(nm) tivity

Cu 0.1278 FCC 1.9 +2C 0.071H 0.046O 0.060Ag 0.1445 FCC 1.9 +1Al 0.1431 FCC 1.5 +3Co 0.1253 HCP 1.8 +2Cr 0.1249 BCC 1.6 +3Fe 0.1241 BCC 1.8 +2Ni 0.1246 FCC 1.8 +2Pd 0.1376 FCC 2.2 +2Zn 0.1332 HCP 1.6 +2