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, c 2. Take reciprocals of intercepts 3. Reduce to smallest integer values 4. Enclose in parentheses, no commas i.e., (hkl)

Crystallographic Planes z x y a b c Example 1 a b c 1. Intercepts 1 1  2. Reciprocals 1/1 1/1 1/ 1 1 0 3. Reduction 1 1 0 4. Miller Indices (110) Example 2 a b c z x y a b c 1. Intercepts ½   2. Reciprocals 1/½ 1/ 1/ 2 0 0 3. Reduction 2 0 0 4. Miller Indices (200) ½

Crystallographic Planes z x y a b c Example 3 a b c 1. Intercepts ½ 1 ¾ 2. Reciprocals 1/½ 1/1 1/¾ 2 1 4/3 3. Reduction 6 3 4 4. Miller Indices (634)

Single Crystals • Single Crystals -Properties vary with E (diagonal) = 273 GPa E (edge) = 125 GPa • Single Crystals -Properties vary with direction: anisotropic. 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.) -Example: the modulus of elasticity (E) in BCC iron:

Fracture surface of failed stainless steel bolt

200X – As cast aluminum from exercise equipment

Polycrystalline Materials Properties may/may not vary with direction. If grains are randomly oriented: isotropic (Epoly iron = 210 GPa) If grains are textured: anisotropic 200 mm

Crystal Defects Types of Defects • Vacancies • Interstitial atoms There is no such thing as a perfect crystal We use/engineer the imperfections to control properties Types of Defects • Vacancies • Interstitial atoms • Substitutional atoms Point defects • Dislocations Linear defects • Grain Boundaries Interfacial defects

Point Defects Vacancy self- interstitial • Vacancies: -vacant atomic sites in a structure. Vacancy distortion of planes • Self-Interstitials: -"extra" atoms positioned between atomic sites. self- interstitial distortion of planes

Equilibrium Concentration: Point Defects • Equilibrium concentration varies with temperature! No. of defects Activation energy æ - ö N Q v ç v ÷ = exp ç ÷ No. of potential è ø N k T defect sites. Temperature Boltzmann's constant -23 (1.38 x 10 J/atom-K) -5 (8.62 x 10 eV/atom-K) k = R/Na Each lattice site Notes: Form is of an Arrehnius Equation PV=nRT; n is the number of moles PV=NkT; N is the number of molecules is a potential vacancy site

Measuring Activation Energy ç ÷ N v = exp - Q k T æ è ö ø • We can get Qv from an experiment. • Measure this... N v T exponential dependence! defect concentration • Replot it... 1/ T N v ln - Q /k slope

Estimating Vacancy Concentration • Find the equil. # of vacancies in 1 m3 of Cu at 1000C. • Given: 3 r = 8.4 g / cm A = 63.5 g/mol Cu Q = 0.9 eV/atom N = 6.02 x 1023 atoms/mol A v 8.62 x 10-5 eV/atom-K 0.9 eV/atom 1273K ç ÷ N v = exp - Q k T æ è ö ø = 2.7 x 10-4 For 1 m3 , N = N A Cu r x 1 m3 = 8.0 x 1028 sites • Answer: N v = (2.7 x 10-4)(8.0 x 1028) sites = 2.2 x 1025 vacancies

Observing Equilibrium Vacancy Conc. • 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 in Solids and the Stability of Surface Morphology", Nature, Vol. 412, pp. 622-625 (2001). Image is 5.75 mm by 5.75 mm.) Copyright (2001) Macmillan Publishers, Ltd. I sland 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

Solid Solutions Two outcomes if impurity (B) added to host (A): OR • Solid solution of B in A (i.e., random dist. of point defects) OR Substitutional solid soln. (e.g., Cu in Ni) Interstitial solid soln. (e.g., C in Fe) • Solid solution of B in A plus particles of a new phase (usually for a larger amount of B) Second phase particle --different composition --often different structure.

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 Rules 1. 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 Valency Metals 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 Rules 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 predict more 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? Element Atomic Crystal Electro- Valence Radius Structure nega- (nm) tivity Cu 0.1278 FCC 1.9 +2 C 0.071 H 0.046 O 0.060 Ag 0.1445 FCC 1.9 +1 Al 0.1431 FCC 1.5 +3 Co 0.1253 HCP 1.8 +2 Cr 0.1249 BCC 1.6 +3 Fe 0.1241 BCC 1.8 +2 Ni 0.1246 FCC 1.8 +2 Pd 0.1376 FCC 2.2 +2 Zn 0.1332 HCP 1.6 +2 Table on p. 106, Callister 7e.