Chapter 3: Structures of Metals & Ceramics

1. Chapter 3: Structures of Metals & Ceramics

2. Structures • 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.

3. Energy and Packing Energy typical neighbor bond length typical neighbor r bond energy • Dense, ordered packing Energy typical neighbor bond length r typical neighbor bond energy • Non dense, random packing Dense, ordered packed structures tend to have lower energies.

4. Materials and Packing Crystalline materials... • atoms pack in periodic, 3D arrays • typical of: -metals -many ceramics -some polymers crystalline SiO2 Si Oxygen Noncrystalline materials... • atoms have no periodic packing • occurs for: -complex structures -rapid cooling "Amorphous" = Noncrystalline noncrystalline SiO2

5.  Metallic Crystal Structures • How can we stack metal atoms to minimize empty space? 2-dimensions vs.

6. Metallic Crystal Structures • 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.

7. Simple Cubic Structure (SC) • Rare due to low packing density (only Po has this structure) • Close-packed directions are cube edges. • Coordination # = 6 (# nearest neighbors)

8. Atomic Packing Factor (APF) volume atoms atom 4 a 3 unit cell p (0.5a) 3 R=0.5a volume close-packed directions unit cell contains 8 x 1/8 = 1 atom/unit cell Volume of atoms in unit cell* APF = Volume of unit cell *assume hard spheres • APF for a simple cubic structure = 0.52 1 APF = 3 a

9. Body Centered Cubic Structure (BCC) • Atoms touch each other along cube diagonals. All atoms are identical. ex: Cr, W, Fe (), Tantalum, Molybdenum • Coordination # = 8 2 atoms/unit cell: 1 center + 8 corners x 1/8

10. Atomic Packing Factor: BCC a 3 a 2 Close-packed directions: R 3 a length = 4R = a atoms volume 4 3 p ( 3 a/4 ) 2 unit cell atom 3 APF = volume 3 a unit cell • APF for a body-centered cubic structure = 0.68 a

11. Face Centered Cubic Structure (FCC) • Atoms touch each other along face diagonals. --Note: All atoms are identical; the face-centered atoms are shaded differently only for ease of viewing. ex: Al, Cu, Au, Pb, Ni, Pt, Ag • Coordination # = 12 4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8

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13. Atomic Packing Factor: FCC 2 a Unit cell contains: 6 x1/2 + 8 x1/8 = 4 atoms/unit cell a atoms volume 4 3 p ( 2 a/4 ) 4 unit cell atom 3 APF = volume 3 a unit cell • APF for a face-centered cubic structure = 0.74 maximum achievable APF Close-packed directions: 2 a length = 4R =

14. Hexagonal Close-Packed Structure (HCP – another view)

15. Hexagonal Close-Packed Structure (HCP) A sites Top layer c Middle layer B sites A sites Bottom layer a • ABAB... Stacking Sequence • 3D Projection • 2D Projection 6 atoms/unit cell • Coordination # = 12 ex: Cd, Mg, Ti, Zn • APF = 0.74 • c/a = 1.633

16. ABAB... Stacking Sequence c03f30

17. X-Ray Diffraction

18. Detector Xray-Tube Xray-Tube Sample Metal Target (Cu or Co) Detector X-Ray Diffractometer sample d – distance between the same atomic planes λ – monochromatic wavelength θ – angle of diffracto- meter Bragg´s Equation d = λ/2 sinθ

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20. nA VCNA Mass of Atoms in Unit Cell Total Volume of Unit Cell  = Theoretical Density, r 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/mol

21. R a 2 52.00 atoms g  = unit cell mol atoms 3 a 6.022x1023 mol volume unit cell Theoretical Density, r • Ex: Cr (BCC) A(atomic weight) =52.00 g/mol n = 2 atoms/unit cell R = 0.125 nm a = 4R/ 3 = 0.2887 nm theoretical = 7.18 g/cm3 ractual = 7.19 g/cm3

22. - - - - - - + + + - - - - - - unstable - F 2+ Ca + CaF : 2 anions cation - F A X m p m, p values to achieve charge neutrality Factors that Determine Crystal Structure 1.Relative sizes of ions – Formation of stable structures: --maximize the # of oppositely charged ion neighbors. stable stable 2.Maintenance of Charge Neutrality: --Net charge in ceramic should be zero. --Reflected in chemical formula:

23. Atomic Bonding in Ceramics CaF2: large SiC: small • 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:

24. Coordination # and Ionic Radii r cation r anion r ZnS cation (zinc blende) r anion NaCl (sodium chloride) CsCl (cesium chloride) • Coordination # increases with To form a stable structure, how many anions can surround a cation? Coord # linear < 0.155 2 triangular 0.155 - 0.225 3 tetrahedral 0.225 - 0.414 4 octahedral 0.414 - 0.732 6 cubic 0.732 - 1.0 8

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26. Computation of Minimum Cation-Anion Radius Ratio a • Determine minimum rcation/ranionfor an octahedral site (C.N. = 6) Measure the radii (blue and yellow spheres) a= 2ranion Substitute for “a” in the above equation

27. Example Problem:Predicting the Crystal Structure of FeO • Answer: Cation Ionic radius (nm) 3+ Al 0.053 2 + Fe 0.077 3+ Fe 0.069 2+ Ca 0.100 based on this ratio, -- coord # = 6 because 0.414 < 0.550 < 0.732 -- crystal structure is similar to NaCl Anion 2- O 0.140 - Cl 0.181 - F 0.133 • On the basis of ionic radii, what crystal structure would you predict for FeO?

28. Rock Salt Structure Same concepts can be applied to ionic solids in general. Example: NaCl (rock salt) structure rNa = 0.102 nm rCl = 0.181 nm • rNa/rCl = 0.564 • cations (Na+) prefer octahedralsites

29. MgO and FeO MgO and FeO also have the NaCl structure O2- rO = 0.140 nm Mg2+ rMg = 0.072 nm • rMg/rO = 0.514 • cations prefer octahedral sites So each Mg2+ (or Fe2+) has 6 neighbor oxygen atoms

30. AX Crystal Structures AX–Type Crystal Structures include NaCl, CsCl, and zinc blende Cesium Chloride structure:  Since 0.732 < 0.939 < 1.0, cubicsites preferred So each Cs+ has 8 neighbor Cl-

31. AX2 Crystal Structures Fluorite structure • Calcium Fluorite (CaF2) • Cations in cubic sites • UO2, ThO2, ZrO2, CeO2 • Antifluorite structure – • positions of cations and anions reversed

32. ABX3 Crystal Structures • Perovskite structure • Ex: complex oxide • BaTiO3

33. SUMMARY • Atoms may assemble into crystalline or amorphous structures. • 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. • We can predict the density of a material, provided we know the atomic weight, atomic radius, and crystal geometry (e.g., FCC, BCC, HCP). • Interatomic bonding in ceramics is ionic and/or covalent. • Ceramic crystal structures are based on: -- maintaining charge neutrality -- cation-anion radii ratios.