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Unit cell/ packing efficiency

Unit cell/ packing efficiency. Given 8 spheres to stack, how would you do it?. Simple cubic structure. Coordination Polyhedra. Consider coordination of anions about a central cation. Halite. Na. Cl. Cl. Cl. Cl. Coordination Polyhedra. Na. Could do the opposite, but conventionally

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Unit cell/ packing efficiency

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  1. Unit cell/ packing efficiency

  2. Given 8 spheres to stack, how would you do it? • Simple cubic structure

  3. Coordination Polyhedra • Consider coordination of anions about a central cation Halite Na Cl Cl Cl Cl

  4. Coordination Polyhedra Na • Could do the opposite, but conventionally choose the cation • Can predict the coordination by considering the radius ratio: RC/RA Cations are generally smaller than anions so begin with maximum ratio = 1.0 Na Na Cl Na

  5. Coordination Polyhedra Radius Ratio: RC/RA = 1.0 (commonly native elements) • Equal sized spheres • “Closest Packed” • Hexagonal array: • 6 nearest neighbors in the plane • Note dimples in which next layer atoms will settle • Two dimple types: • Type 1 point NE • Type 2 point SW • They are equivalent since you could rotate the whole structure 60o and exchange them 2 1

  6. Closest Packing • Add next layer (red) • Red atoms can only settle in one dimple type • Both types are identical and red atoms could settle in either • Once first red atom settles in, can only fill other dimples of that type • In this case filled all type 2 dimples 1

  7. Closest Packing • Third layer ?? • Third layer dimples are now different! • Call layer 1 A sites • Layer 2 = B sites (no matter which choice of dimples is occupied) • Layer 3 can now occupy A-type site (directly above yellow atoms) or C-type site (above voids in both A and B layers)

  8. Closest Packing • Third layer: • If occupy A-type site the layer ordering becomes A-B-A-B and creates a hexagonalclosest packedstructure(HCP) • Coordination number (nearest or touching neighbors) = 12 • 6 coplanar • 3 above the plane • 3 below the plane

  9. Closest Packing • Third layer: • If occupy A-type site the layer ordering becomes A-B-A-B and creates a hexagonalclosest packedstructure(HCP)

  10. Closest Packing • Third layer: • If occupy A-type site the layer ordering becomes A-B-A-B and creates a hexagonalclosest packedstructure(HCP)

  11. Closest Packing • Third layer: • If occupy A-type site the layer ordering becomes A-B-A-B and creates a hexagonalclosest packedstructure(HCP)

  12. Closest Packing • Third layer: • If occupy A-type site the layer ordering becomes A-B-A-B and creates a hexagonalclosest packedstructure(HCP) • Note top layer atoms are directly above bottom layer atoms

  13. Closest Packing • Third layer: • Unit cell

  14. Closest Packing • Third layer: • Unit cell

  15. Closest Packing • Third layer: • Unit cell

  16. Closest Packing • Third layer: • View from top shows hexagonal unit cell

  17. Closest Packing • Third layer: • View from top shows hexagonal unit cell • Mg is HCP

  18. Closest Packing • Alternatively we could place the third layer in the C-type site (above voids in both A and B layers)

  19. Closest Packing • Third layer: • If occupy C-type site the layer ordering is A-B-C-A-B-C and creates a cubicclosest packedstructure(CCP) • Blue layer atoms are now in a unique position above voids between atoms in layers A and B

  20. Closest Packing • Third layer: • If occupy C-type site the layer ordering is A-B-C-A-B-C and creates a cubicclosest packedstructure(CCP) • Blue layer atoms are now in a unique position above voids between atoms in layers A and B

  21. Closest Packing • Third layer: • If occupy C-type site the layer ordering is A-B-C-A-B-C and creates a cubicclosest packedstructure(CCP) • Blue layer atoms are now in a unique position above voids between atoms in layers A and B

  22. Closest Packing • Third layer: • If occupy C-type site the layer ordering is A-B-C-A-B-C and creates a cubicclosest packedstructure(CCP) • Blue layer atoms are now in a unique position above voids between atoms in layers A and B

  23. Closest Packing • Third layer: • If occupy C-type site the layer ordering is A-B-C-A-B-C and creates a cubicclosest packedstructure(CCP) • Blue layer atoms are now in a unique position above voids between atoms in layers A and B

  24. Closest Packing • View from the same side shows the face-centered cubic unit cell that results. • The atoms are slightly shrunken to aid in visualizing the structure A-layer C-layer B-layer A-layer

  25. Closest Packing • Rotating toward a top view

  26. Closest Packing • Rotating toward a top view

  27. Closest Packing • You are looking at a top yellow layer A with a blue layer C below, then a red layer B and a yellow layer A again at the bottom

  28. Closest Packing • CCP is same as face centered cubic • Al is CCP

  29. What happens when RC/RA decreases? • The center cation becomes too small for the site (as if a hard-sphere atom model began to rattle in the site) and it drops to the next lower coordination number (next smaller site). • It will do this even if it is slightly too large for the next lower site. • It is as though it is better to fit a slightly large cation into a smaller site than to have one rattle about in a site that is too large.

  30. The next smaller crystal site is: • Body-Centered Cubic (BCC) with cation (red) in the center of a cube • All cations need to be the same element for BCC • Coordination number is now 8 (corners of cube)

  31. A central cation will remain in VIII coordination with decreasing RC/RA until it again reaches the limiting situation in which all atoms mutually touch. • Then a hard-sphere cation would “rattle” in the position, and it would shift to the next lower coordination (next smaller site). • What is the RC/RA of that limiting condition?? Set = 1 arbitrary since will deal with ratios Diagonal length then = 2

  32. A central cation will remain in VIII coordination with decreasing RC/RA until it again reaches the limiting situation in which all atoms mutually touch. • Then a hard-sphere cation would “rattle” in the position, and it would shift to the next lower coordination (next smaller site). • What is the RC/RA of that limiting condition?? Rotate

  33. A central cation will remain in VIII coordination with decreasing RC/RA until it again reaches the limiting situation in which all atoms mutually touch. • Then a hard-sphere cation would “rattle” in the position, and it would shift to the next lower coordination (next smaller site). • What is the RC/RA of that limiting condition?? Rotate

  34. A central cation will remain in VIII coordination with decreasing RC/RA until it again reaches the limiting situation in which all atoms mutually touch. • Then a hard-sphere cation would “rattle” in the position, and it would shift to the next lower coordination (next smaller site). • What is the RC/RA of that limiting condition?? Rotate

  35. A central cation will remain in VIII coordination with decreasing RC/RA until it again reaches the limiting situation in which all atoms mutually touch. • Then a hard-sphere cation would “rattle” in the position, and it would shift to the next lower coordination (next smaller site). • What is the RC/RA of that limiting condition?? Rotate

  36. A central cation will remain in VIII coordination with decreasing RC/RA until it again reaches the limiting situation in which all atoms mutually touch. • Then a hard-sphere cation would “rattle” in the position, and it would shift to the next lower coordination (next smaller site). • What is the RC/RA of that limiting condition?? Rotate

  37. A central cation will remain in VIII coordination with decreasing RC/RA until it again reaches the limiting situation in which all atoms mutually touch. • Then a hard-sphere cation would “rattle” in the position, and it would shift to the next lower coordination (next smaller site). • What is the RC/RA of that limiting condition?? Rotate

  38. A central cation will remain in VIII coordination with decreasing RC/RA until it again reaches the limiting situation in which all atoms mutually touch. • Fe, Na will form in body centered cubic

  39. CCP coordination = 12 HCP coordination = 12 Body centered coordination = 8 Rc/Ra = 1.0 Rc/Ra = 1.0 Rc/Ra = 0.732 - 1.0 The limits for VIII coordination are thus between 1.0 (when it would by CCP or HCP) and 0.732

  40. As RC/RA continues to decrease below the 0.732 the cation will move to the next lower coordination: VI, or octahedral. The cation is in the center of an octahedron of closest-packed oxygen atoms

  41. As RC/RA continues to decrease below the 0.732 the cation will move to the next lower coordination: VI, or octahedral. The cation is in the center of an octahedron of closest-packed oxygen atoms

  42. As RC/RA continues to decrease below the 0.732 the cation will move to the next lower coordination: VI, or octahedral. The cation is in the center of an octahedron of closest-packed oxygen atoms

  43. As RC/RA continues to decrease below the 0.732 the cation will move to the next lower coordination: VI, or octahedral. The cation is in the center of an octahedron of closest-packed oxygen atoms

  44. As RC/RA continues to decrease below the 0.732 the cation will move to the next lower coordination: VI, or octahedral. The cation is in the center of an octahedron of closest-packed oxygen atoms

  45. As RC/RA continues to decrease below the 0.414 the cation will move to the next lower coordination: IV, or tetrahedral. The cation is in the center of an tetrahedron of closest-packed oxygen atoms

  46. As RC/RA continues to decrease below the 0.414 the cation will move to the next lower coordination: IV, or tetrahedral. The cation is in the center of an tetrahedron of closest-packed oxygen atoms

  47. As RC/RA continues to decrease below the 0.414 the cation will move to the next lower coordination: IV, or tetrahedral. The cation is in the center of an tetrahedron of closest-packed oxygen atoms

  48. As RC/RA continues to decrease below the 0.414 the cation will move to the next lower coordination: IV, or tetrahedral. The cation is in the center of an tetrahedron of closest-packed oxygen atoms

  49. As RC/RA continues to decrease below the 0.414 the cation will move to the next lower coordination: IV, or tetrahedral. The cation is in the center of an tetrahedron of closest-packed oxygen atoms

  50. As RC/RA continues to decrease below the 0.414 the cation will move to the next lower coordination: IV, or tetrahedral. The cation is in the center of an tetrahedron of closest-packed oxygen atoms

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