I. Structural Aspects Sphere PackingsWells, pp Densest Packing of Spheres Two-Dimensions: Unit Cell Hand-Outs: 7

I. Structural Aspects Sphere Packings: GeometryWells, pp Densest Packing of Spheres Two-Dimensions: PERIODIC a a  a = side of unit cell (Å, pm)  = angle between 2 sides =120  “Coordinate System” Unit Cell Hand-Outs: 7

I. Structural Aspects Sphere Packings: GeometryWells, pp Densest Packing of Spheres Two-Dimensions: PERIODIC “A” a a  Unit Cell A: (0, 0) a = side of unit cell (Å, pm)  = angle between 2 sides =120  Hand-Outs: 7

I. Structural Aspects Sphere Packings: GeometryWells, pp Densest Packing of Spheres Two-Dimensions: PERIODIC “A” a a  Unit Cell A: (0, 0) B: (1/3, 2/3) “B” a = side of unit cell (Å, pm)  = angle between 2 sides =120  Hand-Outs: 7

I. Structural Aspects Sphere Packings: GeometryWells, pp Densest Packing of Spheres Two-Dimensions: PERIODIC “A” a a  Unit Cell A: (0, 0) B: (1/3, 2/3) C: (2/3, 1/3) “Fractional Coordinates” “B” “C” a = side of unit cell (Å, pm)  = angle between 2 sides =120  Hand-Outs: 7

I. Structural Aspects Sphere Packings: Closest PackingsWells, pp Densest Packing of Spheres Three-Dimensions: Tetrahedron, ca. 79% Efficiency 70.5  Hand-Outs: 7

I. Structural Aspects Sphere Packings: Closest PackingsWells, pp Densest Packing of Spheres Three-Dimensions: Tetrahedron, ca. 79% Efficiency 70.5  CANNOT fill 3D space with just tetrahedra! Hand-Outs: 7

I. Structural Aspects Sphere Packings: Closest PackingsWells, pp Densest Packing of Spheres Three-Dimensions: Tetrahedron, ca. 79% Efficiency 70.5  CANNOT fill 3D space with just tetrahedra! Three-Dimensions: PERIODIC (Closest Packing: ca. 74% Efficiency) 1 st Layer: over “A” sites (0, 0) 2 nd Layer: over “B” sites (1/3, 2/3) Hand-Outs: 7

I. Structural Aspects Sphere Packings: Closest PackingsWells, pp Hexagonally Closest Packed HCP Cubic Closest Packed CCP = FCC A B A B A B C A  ABAB   h  “h” = “BAB” or “CBC” or …  ABCABC   c  “c” = “ABC” or “BCA” or … Jagodzinski Symbol c Coordination Environments Hand-Outs: 8

I. Structural Aspects Sphere Packings: Closest PackingsWells, pp Hexagonally Closest Packed HCP Cubic Closest Packed CCP = FCC A B A B A B C A  ABAB   h  “h” = “BAB” or “CBC” or … Unit Cell c-axis: 2 closest packed layers c/a =  (8/3) = A: (0, 0, 0); B: (1/3, 2/3, 1/2)  ABCABC   c  “c” = “ABC” or “BCA” or … Unit Cell c-axis: 3 closest packed layers c/a = (  (8/3)(3/2) =  6 =  3 / (1/  2) A: (0, 0, 0); B: (1/3, 2/3, 1/3); C: (2/3, 1/3, 2/3) Jagodzinski Symbol c Hand-Outs: 8

I. Structural Aspects Sphere Packings: Closest PackingsWells, pp La (DHCP): B C A A A  ABACABAC  Sm:  hhc  Examples: Hand-Outs: 8

I. Structural Aspects Sphere Packings: Closest PackingsWells, pp La (DHCP): B C A A A  ABACABAC   hchchc  “  hc  ” Sm:  hhc  =  BABACACBCBAB  Examples: Hand-Outs: 8

Number of CP Layers in Unit Cell Number of Different Sequences Stacking Sequence Jagodzinski Symbol 21  AB  h  31  ABC  c  41  ABAC  hc  I. Structural Aspects Sphere Packings: Closest PackingsWells, pp Exercise: Fill in the Blanks, at Least for 5-8 layers Hand-Outs: 9

I. Structural Aspects Sphere Packings: Packing EfficienciesWells, pp Body-Centered Cubic Packing Unit Cell Efficiency = 2V sphere / V cell Hand-Outs: 10

I. Structural Aspects Sphere Packings: Packing EfficienciesWells, pp Body-Centered Cubic Packing Efficiency = 2V sphere / V cell V cell = a 3 R sphere = (  3/4)a V sphere = (4  /3)(R sphere ) 3 = (  3  /16)a 3 Unit Cell a 2a2a Hand-Outs: 10

I. Structural Aspects Sphere Packings: Packing EfficienciesWells, pp CNNameSphere Density 6Simple Cubic Simple Hexagonal Body-Centered Cubic Body-Centered Tetragonal Tetragonal Close-Packing Closest Packing Body-Centered Cubic Packing Unit Cell a 2a2a Coordination Environment Efficiency = 2V sphere / V cell = (  3  /8) = V cell = a 3 R sphere = (  3/4)a V sphere = (4  /3)(R sphere ) 3 = (  3  /16)a 3 Hand-Outs: 10

I. Structural Aspects Sphere Packings: Interstitial SitesWells, pp How to Quickly Draw a Closest Packing: Projection of 2 closest packed planes Hand-Outs: 11

I. Structural Aspects Sphere Packings: Interstitial SitesWells, pp How to Quickly Draw a Closest Packing: Hand-Outs: 11

I. Structural Aspects Sphere Packings: Interstitial SitesWells, pp How to Quickly Draw a Closest Packing: Hand-Outs: 11

I. Structural Aspects Sphere Packings: Interstitial SitesWells, pp Octahedral “Holes” (Voids): 2 closest packed layers: 1 octahedral void / 2 atoms  closest packed layers : 1 octahedral void / 1 atom Hand-Outs: 11

I. Structural Aspects Sphere Packings: Interstitial SitesWells, pp Octahedral “Holes” (Voids): 2 closest packed layers: 1 octahedral void / 2 atoms  closest packed layers : 1 octahedral void / 1 atom HCP: share faces, edges  AcBcAcBc  CCP: share edges, corners  AcBaCbAcBaCb  A B c Hand-Outs: 11

I. Structural Aspects Sphere Packings: Interstitial SitesWells, pp Tetrahedral “Holes” (Voids): 2 closest packed layers: 2 tetrahedral voids / 2 atoms  closest packed layers : 2 tetrahedral void / 1 atom Hand-Outs: 11

I. Structural Aspects Sphere Packings: Interstitial SitesWells, pp Tetrahedral “Holes” (Voids): 2 closest packed layers: 2 tetrahedral voids / 2 atoms  closest packed layers : 2 tetrahedral void / 1 atom HCP: share faces, edges  AbaBabAbaBab  CCP: share edges, corners  AbaBcbCacAbaBcbCac  A B a b Hand-Outs: 11

I. Structural Aspects Sphere Packings: Radius RatiosWells, pp Coordination Number Optimum Radius Ratio Coordination Polyhedron Tetrahedron Octahedron 0.528Trigonal Prism Cube Tricapped Trigonal Prism Icosahedron 1.000Cuboctahedron (ccp) 1.000Triangular Orthobicupola (hcp) Octahedral Hole Hand-Outs: 11

I. Structural Aspects Atomic and Ionic Sizes Wells, pp , , Atoms and ions are not “hard spheres;” What factors inflence “atomic radii”? Hand-Outs: 12

Atoms and ions are not “hard spheres;” What factors inflence “atomic radii”? (1) Repulsive Forces: approach of uncharged atoms with filled valence subshells (van der Waals radii) (2) Attractive Forces: effective nuclear charge; orbital overlap; electrostatic (metallic, covalent or ionic radii) I. Structural Aspects Atomic and Ionic Sizes Wells, pp , , Hand-Outs: 12

Atoms and ions are not “hard spheres;” What factors inflence “atomic radii”? (1) Repulsive Forces: approach of uncharged atoms with filled valence subshells (van der Waals radii) (2) Attractive Forces: effective nuclear charge; orbital overlap; electrostatic (metallic, covalent or ionic radii) Scales of Atomic and Ionic Radii: Slater, Goldschmidt, Pauling – empirical, based on extensive surveys of interatomic distances. Some corrected for coordination numbers, ionicity, valence bond types, etc. I. Structural Aspects Atomic and Ionic Sizes Wells, pp , , Hand-Outs: 12

Metallic Radii: CN = 12 For ideal cp structures (CCP, HCP with c/a = 1.63):R 12 = d / 2 For distorted cp structures:R 12 =  d  / 2 For lower CN: Relative Metallic Radii (Goldschmidt) CN = 8:R 8 = 0.97 R 12 CN = 6:R 6 = 0.96 R 12 CN = 4:R 4 = 0.88 R 12 I. Structural Aspects Atomic and Ionic Sizes Wells, pp , , Hand-Outs: 12

M(s)  M(g) Filling M-M Bonding States Filling M-M Antibonding States Minimum Radii Maximum Cohesive E. I. Structural Aspects Atomic and Ionic Sizes Wells, pp , , Hand-Outs: 13

Metallic Radii: CN = 12 For ideal cp structures (CCP, HCP with c/a = 1.63):R 12 = d / 2 For distorted cp structures:R 12 =  d  / 2 For lower CN: Relative Metallic Radii (Goldschmidt) CN = 8:R 8 = 0.97 R 12 CN = 6:R 6 = 0.96 R 12 CN = 4:R 4 = 0.88 R 12 Estimation Strategies: (1) Constant V atom (How to estimate R 12 from BCC elements (R 8 )) FCC vs. BCC:V atom = (a FCC ) 3 / 4 = (a BCC ) 3 / 2 d FCC = 2R 12 = a FCC /  2 d BCC = 2R 8 =  3 a BCC / 2 Therefore, R 8 = R 12 I. Structural Aspects Atomic and Ionic Sizes Wells, pp , , Hand-Outs: 12-13

Metallic Radii: CN = 12 For ideal cp structures (CCP, HCP with c/a = 1.63):R 12 = d / 2 For distorted cp structures:R 12 =  d  / 2 For lower CN: Relative Metallic Radii (Goldschmidt) CN = 8:R 8 = 0.97 R 12 CN = 6:R 6 = 0.96 R 12 CN = 4:R 4 = 0.88 R 12 Estimation Strategies: (1) Constant V atom (2) Use alloys that show close packed structures, e.g., Ag 3 Sb (HCP) – provides R 12 (Sb) I. Structural Aspects Atomic and Ionic Sizes Wells, pp , , Hand-Outs: 12-13

Metallic Radii: CN = 12 For ideal cp structures (CCP, HCP with c/a = 1.63):R 12 = d / 2 For distorted cp structures:R 12 =  d  / 2 For lower CN: Relative Metallic Radii (Goldschmidt) CN = 8:R 8 = 0.97 R 12 CN = 6:R 6 = 0.96 R 12 CN = 4:R 4 = 0.88 R 12 Estimation Strategies: (1) Constant V atom (2) Use alloys that show close packed structures, e.g., Ag 3 Sb (HCP) – provides R 12 (Sb) (3) Linear extrapolation of solid solutions of the element in a close packed metal. I. Structural Aspects Atomic and Ionic Sizes Wells, pp , , Hand-Outs: 12-13