Chapter 3 -1 ISSUES TO ADDRESS... What is the difference in atomic arrangement between crystalline and noncrystalline solids? What features of a metal’s/ceramic’s atomic structure determine its density? Under what circumstances does a material property vary with the measurement direction? Chapter 3: Structures of Metals & Ceramics How do the crystal structures of ceramic materials differ from those for metals?

Chapter 3 -2 Non dense, ________________ Dense, ___________________ Energy and Packing Energy r _______________ bond length typical neighbor bond energy Energy r typical neighbor bond length typical neighbor bond energy Dense, _____________________________ to have _________ energies.

Chapter 3 -3 Crystalline materials... atoms have no periodic packing Noncrystalline materials... crystalline SiO 2 noncrystalline SiO 2 "Amorphous" = Noncrystalline Adapted from Fig. 3.41(b), Callister & Rethwisch 4e. Adapted from Fig. 3.41(a), Callister & Rethwisch 4e. Materials and Packing SiOxygen typical of: occurs for: atoms pack __________, 3D arrays -________ -many _________ -some _________ - _________________ - _____________

Chapter 3 -4 Metallic Crystal Structures How can we stack metal atoms to minimize empty space? 2-dimensions vs. Now stack these 2-D layers to make 3-D structures

Chapter 3 -5 Tend to be densely packed. Reasons for dense packing: - Typically, only one __________is present, so all atomic ________ are the same. - Metallic bonding is not ________________. - Nearest neighbor distances tend to be small in order to __________ bond energy. - ______________ cloud shields cores from each other We will examine three such structures... Metallic Crystal Structures Metals have the simplest crystal structures.

Chapter 3 -6 Rare due to low packing density (only Po has this structure) Close-packed ___________ are cube edges. Coordination # = ___ (# nearest neighbors) Simple Cubic Structure (SC) Click once on image to start animation (Courtesy P.M. Anderson)

Chapter 3 -7 APF for a simple _______ structure = 0.52 APF = a  (0.5a) 3 1 atoms unit cell atom volume unit cell volume Atomic Packing Factor (APF) APF = Volume of atoms in unit cell* Volume of unit cell *assume hard spheres Adapted from Fig. 3.43, Callister & Rethwisch 4e. close-packed ___________ a R=0.5a contains 8 x 1/8 = 1atom/unit cell

Chapter 3 -8 Coordination # = 8 Adapted from Fig. 3.2, Callister & Rethwisch 4e. Atoms touch each other along cube ____________. --Note: All atoms are identical; the center atom is shaded differently only for ease of viewing. Body Centered Cubic Structure (BCC) ex: Cr, W, Fe (  ), Tantalum, ____________ 2 atoms/unit cell: 1 center + 8 _____________ Click once on image to start animation (Courtesy P.M. Anderson)

Chapter 3 - VMSE Screenshot – BCC Unit Cell 9

Chapter Atomic Packing Factor: BCC a APF = 4 3  (3a/4) 3 2 atoms unit cell atom volume a 3 unit cell volume length = 4R = Close-packed __________: 3 a APF for a body-centered _______ structure = 0.68 a R Adapted from Fig. 3.2(a), Callister & Rethwisch 4e. a 2 a 3

Chapter Coordination # = ___ Adapted from Fig. 3.1, Callister & Rethwisch 4e. Atoms touch each other along face ___________. --Note: All atoms are ____________; the face-centered atoms are shaded differently only for ease of viewing. Face Centered Cubic Structure (FCC) ex: Al, Cu, Au, Pb, Ni, Pt, Ag 4 atoms/unit cell: 6 __________ + 8 corners x 1/8 Click once on image to start animation (Courtesy P.M. Anderson)

Chapter APF for a face-centered cubic structure = 0.74 Atomic Packing Factor: FCC maximum achievable APF Close-packed directions: length = 4R = 2 a a Adapted from Fig. 3.1(a), Callister & Rethwisch 4e. APF = 4 3  (2a/4) 3 atoms unit cell atom volume unit cell volume Unit cell contains: __________________ =

Chapter A sites B B B B B BB C sites C C C A B B ABCABC... __________ Sequence 2D Projection FCC _____ Cell FCC Stacking Sequence B B B B B BB B sites C C C A C C C A

Chapter Coordination # = ___ ABAB... Stacking Sequence APF = _______ 3D Projection 2D Projection Adapted from Fig. 3.3(a), Callister & Rethwisch 4e. Hexagonal Close-Packed Structure (HCP) __ atoms/unit cell ex: Cd, Mg, Ti, Zn c/a = ________ c a A sites B sites A sites Bottom layer Middle layer Top layer

Chapter 3 - VMSE Screenshot – Stacking Sequence and Unit Cell for HCP 15

Chapter Theoretical Density,  Density =  = VC NAVC NA n An A  = Cell Unit of VolumeTotal Cell Unit in Atomsof Mass where n = ________________________ A = atomic weight V C = ________________ = ____________ N A = Avogadro’s number = x atoms/mol

Chapter Ex: Cr (BCC) A = g/mol R = ___________ n = 2 atoms/unit cell  theoretical a = 4R/ 3 = nm  actual a R  = a3a atoms unit cell mol g unit cell volume atoms mol x Theoretical Density,  = 7.18 g/cm 3 = 7.19 g/cm 3 Adapted from Fig. 3.2(a), Callister & Rethwisch 4e.

Chapter Adapted from Fig. 2.7, Callister & Rethwisch 4e. (Fig. 2.7 is adapted from Linus Pauling, The Nature of the Chemical Bond, 3rd edition, Copyright 1939 and 1940, 3rd edition. Copyright 1960 by Cornell University.) Degree of ionic character may be large or small: Atomic Bonding in Ceramics SiC: small CaF 2 : large Bonding: -- ______________________________________. -- % ionic character __________ with difference in electronegativity of atoms.

Chapter Ceramic Crystal Structures Oxide structures –oxygen anions ________ than metal cations –close _______ oxygen in a ______ (usually ____) –cations fit into _______ sites among _______ ions

Chapter Factors that Determine Crystal Structure Adapted from Fig. 3.4, Callister & Rethwisch 4e stable 2. Maintenance of Charge Neutrality : --_________________ should be zero. --Reflected in chemical formula: CaF 2 : Ca 2+ cation F - F - anions + A m X p m, p values to achieve charge neutrality 1. Relative sizes of ions – _______________________: --maximize the # of ___________________________ _________ _______

Chapter __________ # increases with Coordination # and Ionic Radii Adapted from Table 3.3, Callister & Rethwisch 4e. 2 r cation r anion Coord # < linear ________ tetrahedral octahedral _____ Adapted from Fig. 3.5, Callister & Rethwisch 4e. Adapted from Fig. 3.6, Callister & Rethwisch 4e. Adapted from Fig. 3.7, Callister & Rethwisch 4e. ZnS (zinc blende) NaCl (sodium chloride) CsCl (cesium chloride) r cation r anion To form a ________ structure, how many anions can surround around a cation?

Chapter Computation of Minimum Cation-Anion Radius Ratio Determine ________ r cation /r anion for an octahedral site (C.N. = __) _________

Chapter Bond Hybridization Bond Hybridization is possible when there is significant _________ bonding –____________________________ –For example for SiC X Si = 1.8 and X C = 2.5 ~ 89% ___________ bonding Both Si and C prefer sp 3 hybridization Therefore, for SiC, Si atoms occupy ______________ sites

Chapter On the basis of ionic radii, what ________________ would you predict for FeO? Answer: based on this ratio, -- coord # = ___ because < < crystal structure is ____ Data from Table 3.4, Callister & Rethwisch 4e. Example Problem: Predicting the Crystal Structure of FeO Ionic radius (nm) Cation Anion Al 3+ Fe Ca 2+ O 2- Cl - F -

Chapter Rock Salt Structure Same concepts can be applied to ____ solids in general. Example: NaCl (rock salt) structure r Na = nm r Na /r Cl = ________  cations (Na + ) prefer _________ sites Adapted from Fig. 3.5, Callister & Rethwisch 4e. r Cl = _____ nm

Chapter MgO and FeO O 2- r O = nm Mg 2+ r Mg = nm So each Mg 2+ (or Fe 2+ ) _______ neighbor oxygen atoms MgO and FeO also have the NaCl structure r Mg /r O = _______  _________ prefer octahedral sites Adapted from Fig. 3.5, Callister & Rethwisch 4e.

Chapter AX Crystal Structures Cesium Chloride structure: AX–Type Crystal Structures include NaCl, CsCl, and zinc blende  Since < ____ < 1.0, _____ sites preferred So each Cs + has __ neighbor Cl - _____ Adapted from Fig. 3.6, Callister & Rethwisch 4e.

Chapter AX 2 Crystal Structures Fluorite structure Calcium _______ (CaF 2 ) Cations in _______ sites UO 2, ThO 2, ZrO 2, CeO 2 __________ structure – positions of cations and anions reversed Adapted from Fig. 3.8, Callister & Rethwisch 4e.

Chapter ABX 3 Crystal Structures _________ structure Ex: complex oxide _________ Adapted from Fig. 3.9, Callister & Rethwisch 4e.

Chapter 3 - VMSE Screenshot – Zinc Blende Unit Cell 30

Chapter Density Computations for Ceramics Number of formula units/unit cell Volume of unit cell __________ number = sum of atomic weights of _______________________

Chapter Densities of Material Classes  metals _  ceramics _  polymers Why? Data from Table B.1, Callister & Rethwisch, 8e.  (g/cm ) 3 Graphite/ Ceramics/ Semicond Metals/ Alloys Composites/ fibers Polymers Based on data in Table B1, Callister *GFRE, CFRE, & AFRE are Glass, Carbon, & Aramid Fiber-Reinforced Epoxy composites (values based on 60% volume fraction of aligned fibers in an epoxy matrix) Magnesium Aluminum Steels Titanium Cu,Ni Tin, Zinc Silver, Mo Tantalum Gold, W Platinum Graphite Silicon Glass-soda Concrete Si nitride Diamond Al oxide Zirconia HDPE, PS PP, LDPE PC PTFE PET PVC Silicone Wood AFRE* CFRE* GFRE* Glass fibers Carbonfibers Aramid fibers Metals have... close-packing (metallic bonding) often _____atomic masses Ceramics have... less _______packing often lighter elements Polymers have... low packing _________ (often _____________) lighter elements (C,H,O) Composites have... ______________values In general

Chapter Silicate Ceramics Most common __________________________ SiO 2 (silica) _______________ forms are quartz, crystobalite, & tridymite The strong Si-O bonds lead to a high ____________ temperature (1710ºC) for this material Si 4+ O 2- crystobalite Adapted from Figs , Callister & Rethwisch 4e

Chapter Silicates Mg 2 SiO 4 Ca 2 MgSi 2 O 7 Adapted from Fig. 3.12, Callister & Rethwisch 4e. Bonding of adjacent SiO 4 4- accomplished by the sharing of common ___________________ Presence of cations such as Ca 2+, Mg 2+, & Al maintain charge ___________, and 2. ________ bond SiO 4 4- to one another

Chapter Quartz is _____________ SiO 2 : Basic Unit: Glass is ____________(__________) ____________ is SiO 2 to which no impurities have been added Other common _________ contain impurity ions such as Na +, Ca 2+, Al 3+, and B 3+ (soda glass) Adapted from Fig. 3.42, Callister & Rethwisch 4e. Glass Structure Si0 4 tetrahedron 4- Si 4+ O 2- Si 4+ Na + O 2-

Chapter Layered Silicates Layered ______ (e.g., clays, mica, talc) –SiO 4 ___________ connected together to form 2-D plane A net negative charge is associated with each (Si 2 O 5 ) 2- unit Negative charge balanced by _______ plane rich in positively charged _________ Adapted from Fig. 3.13, Callister & Rethwisch 4e.

Chapter Kaolinite clay _________ (Si 2 O 5 ) 2- layer with Al 2 (OH) 4 2+ layer Layered Silicates (cont.) Note: Adjacent sheets of this type _________ bound to one another by ________________________. Adapted from Fig. 3.14, Callister & Rethwisch 4e.

Chapter Polymorphic Forms of Carbon Diamond –tetrahedral bonding of carbon ______________________ ______________________ conductivity –large single crystals – gem stones –small ________ – used to grind/cut other materials –__________ thin films hard surface coatings – used for cutting tools, medical devices, etc. Adapted from Fig. 3.16, Callister & Rethwisch 4e.

Chapter Polymorphic Forms of Carbon (cont) __________ –______ structure – parallel ___________ arrays of carbon atoms –weak van der Waal’s forces between layers –planes slide easily over one another -- good lubricant Adapted from Fig. 3.17, Callister & Rethwisch 4e.

Chapter Polymorphic Forms of Carbon (cont) Fullerenes and Nanotubes ___________ – spherical cluster of 60 carbon atoms, C 60 –Like a soccer ball Carbon __________ – sheet of graphite rolled into a tube –Ends capped with fullerene _______________ Adapted from Figs & 3.19, Callister & Rethwisch 4e.

Chapter Some engineering applications require ________crystals: Properties of __________materials often related to crystal structure. (Courtesy P.M. Anderson) -- Ex: Quartz fractures more easily along some crystal planes than others. -- diamond single crystals for abrasives -- turbine blades Fig. 9.40(c), Callister & Rethwisch 4e. (Fig. 9.40(c) courtesy of Pratt and Whitney). (Courtesy Martin Deakins, GE Superabrasives, Worthington, OH. Used with permission.) Crystals as Building Blocks

Chapter Most engineering materials are ____________. Nb-Hf-W plate with an electron beam weld. Each _______ is a single crystal. If grains are _________ oriented, overall component properties are not ___________. Grain sizes typically range from 1 nm to 2 cm (i.e., from a few to millions of atomic layers). Adapted from Fig. K, color inset pages of Callister 5e. (Fig. K is courtesy of Paul E. Danielson, Teledyne Wah Chang Albany) 1 mm Polycrystals Isotropic __________

Chapter Single Crystals -Properties vary with direction: ____________. -Example: the __________ of elasticity (E) in BCC iron: Data from Table 3.7, Callister & Rethwisch 4e. (Source of data is R.W. Hertzberg, Deformation and Fracture Mechanics of Engineering Materials, 3rd ed., John Wiley and Sons, 1989.) Polycrystals -Properties may/may not vary with direction. -If grains are _________ oriented: __________. (E poly iron = 210 GPa) -If grains are _________, anisotropic. 200  m Adapted from Fig. 5.19(b), Callister & Rethwisch 4e. (Fig. 5.19(b) is courtesy of L.C. Smith and C. Brady, the National Bureau of Standards, Washington, DC [now the National Institute of Standards and Technology, Gaithersburg, MD].) Single vs Polycrystals E (diagonal) = __________ E (edge) = 125 GPa

Chapter Polymorphism Two or more distinct _______ structures for the same material (allotropy/polymorphism) titanium ,  -Ti carbon __________, graphite BCC FCC BCC 1538ºC 1394ºC 912ºC  - Fe  - Fe  - Fe liquid iron system

Chapter Fig. 3.20, Callister & Rethwisch 4e. Crystal Systems 7 crystal systems 14 crystal lattices Unit cell: smallest __________________ which contains the complete ___________ of a crystal. a, b, and c are the ________ constants

Chapter Point Coordinates Point coordinates for ________ center are a/2, b/2, c/2 ½ ½ ½ Point ___________ for unit cell corner are 111 __________: integer multiple of lattice constants  identical position in another unit cell z x y a b c y z 2c2c b b

Chapter Crystallographic Directions ex: ________ of directions z x Algorithm where ___________ represents a negative index _______________ y 1. Vector ___________ (if necessary) to pass through origin. 2. Read off ____________ in terms of unit cell dimensions a, b, and c 3. Adjust to smallest ___________ values 4. Enclose in _______ brackets, no commas [uvw] ___________________________

Chapter 3 - VMSE Screenshot – [101] Direction 48

Chapter Linear Density Linear Density of Atoms  LD = a [110] Unit length of direction vector Number of atoms # atoms length nm a2 LD   Adapted from Fig. 3.1(a), Callister & Rethwisch 4e. ex: linear _________ of Al in [110] direction a = nm

Chapter Drawing HCP Crystallographic Directions (i) 1. Remove ____________ 2. Divide by largest integer so all values are ≤ 1 3. Multiply terms by appropriate unit cell dimension a (for a 1, a 2, and a 3 axes) or c (for z -axis) to produce ________________ 4. Construct _________ by stepping off these ______________ Algorithm (Miller-Bravais coordinates) Adapted from Figure 3.25, Callister & Rethwisch 4e.

Chapter Drawing HCP Crystallographic Directions (ii) Draw the ___________in a hexagonal unit cell. [1213] 4. Construct Vector 1. Remove ________ Algorithm a 1 a 2 a 3 z 2. Divide by 3 3. ____________ proceed – a /3 units along a 1 axis to point p –2 a /3 units parallel to a 2 axis to point q a /3 units parallel to a 3 axis to point r c units parallel to z axis to point s p q r s start at point o Adapted from p. 62, Callister & Rethwisch 8e. [1213] direction represented by vector from point o to point s

Chapter _________________(if necessary) to pass through origin. 2. Read off projections in terms of three- axis ( a 1, a 2, and z ) ___________________ a and c 3. Adjust to smallest ________ values 4. Enclose in square brackets, no commas, for three-axis __________ 5. Convert to four-axis Miller-Bravais lattice coordinates using equations below: 6. Adjust to smallest integer values and enclose in brackets [uvtw] Adapted from p. 74, Callister & Rethwisch 4e. Algorithm Determination of HCP Crystallographic Directions (ii)

Chapter Brackets [110] 1. Reposition not needed 2. Projections a a 0 c Reduction Example a 1 a 2 z 5. Convert to 4-axis parameters 1/3, 1/3, -2/3, 0 => 1, 1, -2, 0 => [ 1120 ] 6. Reduction & Brackets Adapted from p. 74, Callister & Rethwisch 4e. Determination of HCP Crystallographic Directions (ii) Determine indices for green vector

Chapter Crystallographic Planes Adapted from Fig. 3.26, Callister & Rethwisch 4e.

Chapter Crystallographic Planes _____ Indices: Reciprocals of the (three) axial intercepts for a plane, cleared of _________ & common multiples. All ________ planes have same Miller indices. Algorithm 1. Read off ____________ of plane with axes in terms of a, b, c 2. Take ____________ of intercepts 3. Reduce to smallest integer values 4. Enclose in parentheses, no commas i.e., (hkl)

Chapter Crystallographic Planes z x y a b c examplea b c z x y a b c 1. Intercepts 1/2   2. Reciprocals 1/½ 1/  1/  examplea b c 4. Miller Indices _____ 1. Intercepts 1 1 __ 2. Reciprocals 1/1 1/1 ___ Reduction 1 1 __ 3. Reduction __ __ 0

Chapter Crystallographic Planes z x y a b c 4. Miller Indices (634) example 1. Intercepts 1/2 1 3/4 a b c 2. Reciprocals 1/½ 1/1 1/¾ 21 4/3 3. Reduction 63 4 (001)(010), Family of Planes {hkl} (100),(010),(001),Ex: {100} = (100),

Chapter 3 - VMSE Screenshot – Crystallographic Planes 58 Additional practice on indexing crystallographic planes

Chapter Crystallographic Planes (HCP) In hexagonal unit cells the same idea is used example a 1 a 2 a 3 c 4. Miller-Bravais Indices(1011) 1. Intercepts 1  1 2. Reciprocals 1 1/  Reduction1 0 1 a2a2 a3a3 a1a1 z Adapted from Fig. 3.24(b), Callister & Rethwisch 4e.

Chapter Crystallographic Planes We want to examine the ______ packing of crystallographic planes Iron foil can be used as a catalyst. The atomic packing of the exposed _________ is important. a)Draw (100) and (111) crystallographic _____ for Fe. b) Calculate the planar ________ for each of these planes.

Chapter Planar Density of (100) Iron Solution: At T < 912ºC iron has the ________structure. (100) __________of iron R = nm R 3 34 a  Adapted from Fig. 3.2(c), Callister & Rethwisch 4e. 2D repeat unit = Planar Density = a 2 atoms 2D repeat unit = nm 2 atoms ____ m2m2 atoms = ________ 1 2 R 3 34 area 2D repeat unit

Chapter Planar Density of (111) Iron Solution (cont): (___) plane ______ in plane/ unit surface cell R 3 16 R a3ah2area           atoms in plane atoms above plane atoms _____ plane ah 2 3  a 2 2D repeat unit = = nm 2 atoms 7.0 m2m2 atoms __________ 3 2 R 3 16 Planar Density = atoms 2D repeat unit area 2D repeat unit

Chapter 3 - VMSE Screenshot – Atomic Packing – (111) Plane for BCC 63

Chapter X-Ray Diffraction ________ gratings must have spacings comparable to the wavelength of diffracted ___________. Can’t resolve ____________  Spacing is the distance between __________ planes of atoms.

Chapter X-Rays to Determine Crystal Structure X-ray intensity (from detector)   c d d  n 2 sin  c ________________of critical angle,  c, allows computation of planar __________, d. Incoming X-rays _________ from crystal planes. Adapted from Fig. 3.38, Callister & Rethwisch 4e. reflections must be in phase for a detectable signal spacing between _________ d incoming X-rays outgoing X-rays detector   extra distance travelled by wave “2” “1”“1” “2”“2” “1”“1” “2”“2”

Chapter X-Ray Diffraction Pattern Adapted from Fig. 3.40, Callister 4e. (110) (200) (211) z x y a b c Diffraction angle 2  Diffraction pattern for polycrystalline  -iron (BCC) Intensity (relative) z x y a b c z x y a b c

Chapter Atoms may assemble into crystalline or amorphous structures. We can predict the density of a material, provided we know the atomic weight, atomic radius, and crystal geometry (e.g., FCC, BCC, HCP). SUMMARY Common metallic crystal structures are FCC, BCC, and HCP. Coordination number and atomic packing factor are the same for both FCC and HCP crystal structures. Crystallographic points, directions and planes are specified in terms of indexing schemes. Crystallographic directions and planes are related to atomic linear densities and planar densities. Ceramic crystal structures are based on: -- maintaining charge neutrality -- cation-anion radii ratios. Interatomic bonding in ceramics is ionic and/or covalent.

Chapter Some materials can have more than one crystal structure. This is referred to as polymorphism (or allotropy). SUMMARY Materials can be single crystals or polycrystalline. Material properties generally vary with single crystal orientation (i.e., they are anisotropic), but are generally non-directional (i.e., they are isotropic) in polycrystals with randomly oriented grains. X-ray diffraction is used for crystal structure and interplanar spacing determinations.

Chapter Core Problems: Self-help Problems: ANNOUNCEMENTS Reading: