Chapter 3: Structures of Metals & CeramicsStructuresEnergy and PackingMaterials and Packing Metallic Crystal StructuresMetallic Crystal StructuresSimple Cubic Structure (SC)Atomic Packing Factor (APF)Body Centered Cubic Structure (BCC)Atomic Packing Factor: BCCFace Centered Cubic Structure (FCC)c03probAtomic Packing Factor: FCCSlide 14Hexagonal Close-Packed Structure (HCP)c03f30X-Ray DiffractionSlide 18c03tf01Theoretical Density, rSlide 21Factors that Determine Crystal StructureAtomic Bonding in CeramicsCoordination # and Ionic Radiic03tf03Computation of Minimum Cation-Anion Radius RatioExample Problem: Predicting the Crystal Structure of FeORock Salt StructureMgO and FeOAX Crystal StructuresAX2 Crystal StructuresABX3 Crystal StructuresSUMMARYChapter 3: Structures of Metals & Ceramics1Structures•The properties of some materials are directly related to their crystal structures.•Significant property differences exist between crystalline and noncrystalline materials having the same composition.23• Non dense, random packing• Dense, ordered packingDense, ordered packed structures tend to have lower energies.Energy and PackingEnergy rtypical neighbor bond length typical neighbor bond energy Energy rtypical neighbor bond length typical neighbor bond energy4• atoms pack in periodic, 3D arraysCrystalline materials...-metals-many ceramics-some polymers• atoms have no periodic packingNoncrystalline materials...-complex structures-rapid coolingcrystalline SiO2noncrystalline SiO2"Amorphous" = NoncrystallineMaterials and PackingSi Oxygen• typical of:• occurs for:5 Metallic Crystal Structures •How can we stack metal atoms to minimize empty space?2-dimensionsvs.6• Tend to be densely packed.• Reasons for dense packing:-Typically, only one element is present, so all atomic radii are the same.-- Metallic bonding is not directional.-- Nearest neighbor distances tend to be small in order to lower bond energy.- The “electron cloud” shields cores from each other• They have the simplest crystal structures.Metallic Crystal Structures7• Rare due to low packing density (only Po has this structure)• Close-packed directions are cube edges.• Coordination # = 6 (# nearest neighbors)Simple Cubic Structure (SC)8• APF for a simple cubic structure = 0.52APF = a343(0.5a)31atomsunit cellatomvolumeunit cellvolumeAtomic Packing Factor (APF)APF = Volume of atoms in unit cell*Volume of unit cell*assume hard spheresclose-packed directionsaR=0.5acontains 8 x 1/8 = 1 atom/unit cell9• Coordination # = 8• Atoms touch each other along cube diagonals.All atoms are identical. Body Centered Cubic Structure (BCC)ex: Cr, W, Fe (), Tantalum, Molybdenum2 atoms/unit cell: 1 center + 8 corners x 1/810Atomic Packing Factor: BCCaAPF = 43 ( 3a/4)32atomsunit cellatomvolumea3unit cellvolumelength = 4R =Close-packed directions:3 a• APF for a body-centered cubic structure = 0.68aRa 2a 311• Coordination # = 12• Atoms touch each other along face diagonals.--Note: All atoms are identical; the face-centered atoms are shaded differently only for ease of viewing.Face Centered Cubic Structure (FCC)ex: Al, Cu, Au, Pb, Ni, Pt, Ag4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/813• APF for a face-centered cubic structure = 0.74Atomic Packing Factor: FCCmaximum achievable APFAPF = 43( 2a/4)34atomsunit cellatomvolumea3unit cellvolumeClose-packed directions: length = 4R =2 a Unit cell contains: 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cella2 a14Hexagonal Close-Packed Structure (HCP – another view)15• Coordination # = 12• ABAB... Stacking Sequence• APF = 0.74• 3D Projection • 2D ProjectionHexagonal Close-Packed Structure (HCP)6 atoms/unit cellex: Cd, Mg, Ti, Zn• c/a = 1.633caA sitesB sitesA sitesBottom layerMiddle layerTop layerABAB... Stacking SequenceX-Ray Diffraction17Xray-TubeDetectorSampleBragg´s Equationd = λ/2 sinθ d – distance between the same atomic planesλ – monochromatic wavelengthθ – angle of diffracto- meterXray-TubeDetectorMetal Target(Cu or Co)sampleX-Ray Diffractometer20Theoretical Density, where n = number of atoms/unit cell A = atomic weight VC = Volume of unit cell = a3 for cubic NA = Avogadro’s number = 6.022 x 1023 atoms/molDensity =  =VC NAn A =Cell Unit of VolumeTotalCell Unit in Atomsof Mass21•Ex: Cr (BCC) A (atomic weight) = 52.00 g/mol n = 2 atoms/unit cellR = 0.125 nm theoreticala = 4R/ 3 = 0.2887 nmactualaR = a352.002atomsunit cellmolgunit cellvolumeatomsmol6.022 x 1023Theoretical Density, = 7.18 g/cm3= 7.19 g/cm322Factors that Determine Crystal Structure1. Relative sizes of ions – Formation of stable structures: --maximize the # of oppositely charged ion neighbors.--- -+unstable----+stable----+stable2. Maintenance of Charge Neutrality : --Net charge in ceramic should be zero. --Reflected in chemical formula:CaF2:Ca2+cationF-F-anions+AmXpm, p values to achieve charge neutrality23• Bonding: -- Can be ionic and/or covalent in character. -- % ionic character increases with difference in electronegativity of atoms.• Degree of ionic character may be large or small:Atomic Bonding in CeramicsSiC: smallCaF2: large24• Coordination # increases withCoordination # and Ionic Radii2 rcationranionCoord #< 0.155 0.155 - 0.225 0.225 - 0.4140.414 - 0.732 0.732 - 1.03 468lineartriangulartetrahedraloctahedralcubicZnS (zinc blende)NaCl(sodium chloride)CsCl(cesium chloride)rcationranionTo form a stable structure, how many anions can surround a cation?26Computation of Minimum Cation-Anion Radius Ratio•Determine minimum rcation/ranion for an octahedral site (C.N. = 6)a  2ranion 2ranion 2rcation2 2ranion ranion rcation 2ranion rcation( 2  1)ranionarr 222cationanion414.012anioncationrra Measure the radii (blue and yellow spheres)Substitute for “a” in the above equation27• On the basis of ionic radii, what crystal structure would you predict for FeO? • Answer:550014000770anioncation...rrbased on this ratio,-- coord # = 6 because 0.414 < 0.550 < 0.732-- crystal structure is similar to NaClExample Problem: Predicting the Crystal Structure of FeOIonic radius (nm)0.0530.0770.0690.1000.1400.1810.133CationAnionAl3+Fe2+Fe3+Ca2+O2-Cl-F-28Rock Salt StructureSame concepts can be applied to ionic solids in general. Example: NaCl (rock salt) structurerNa = 0.102 nmrNa/rCl