1. l Consider coordination of anions about a central cation Coordination Polyhedra Halite Cl Cl Cl Cl Na

2. l Could do the opposite, but conventionally choose the cation l Can predict the coordination by considering the radius ratio: by considering the radius ratio: R C /R A Cations are generally smaller than anions so begin with maximum ratio = 1.0 Coordination Polyhedra Na Cl

3. Coordination Polyhedra Radius Ratio: R C /R A = 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 1 point NE Type 2 point SW Type 2 point SW They are equivalent since you could rotate the whole structure 60 o and exchange them 1 2

4. 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

5. 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)

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

7. Closest Packing Third layer: If occupy A-type site the layer ordering becomes A-B-A-B and creates a hexagonal closest packed structure (HCP)

8. Closest Packing Third layer: If occupy A-type site the layer ordering becomes A-B-A-B and creates a hexagonal closest packed structure (HCP)

9. Closest Packing Third layer: If occupy A-type site the layer ordering becomes A-B-A-B and creates a hexagonal closest packed structure (HCP)

10. Closest Packing Third layer: If occupy A-type site the layer ordering becomes A-B-A-B and creates a hexagonal closest packed structure (HCP) Note top layer atoms are directly above bottom layer atoms

11. Closest Packing Third layer: Unit cell

12. Closest Packing Third layer: Unit cell

13. Closest Packing Third layer: Unit cell

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

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

16. Hexagonal Closest Packing Click to run animation Case Klein animation for Mineral Science, © John Wiley & Sons

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

18. Closest Packing Third layer: If occupy C-type site the layer ordering is A-B-C-A-B-C and creates a cubic closest packed structure (CCP) Blue layer atoms are now in a unique position above voids between atoms in layers A and B

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

23. 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 B-layer C-layer A-layer

24. Closest Packing Rotating toward a top view

25. Closest Packing Rotating toward a top view

26. 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

27. Cubic Closest Packing Click to run animation Case Klein animation for Mineral Science, © John Wiley & Sons

28. What happens when R C /R A decreases? The center cation becomes too small for the XII site (as if a hard-sphere atom model began to rattle in the XII 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.

29. Coordination Polyhedra Click to run animation Case Klein animation for Mineral Science, © John Wiley & Sons

30. Body-Centered Cubic (BCC) with cation (red) in the center of a cube Coordination number is now 8 (corners of cube) The next smaller crystal site is:

31. 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 R C /R A of that limiting condition?? A central cation will remain in VIII coordination with decreasing R C /R A until it again reaches the limiting situation in which all atoms mutually touch. Set = 1 Diagonal length then = 2 arbitrary since will deal with ratios

32. 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 R C /R A of that limiting condition?? A central cation will remain in VIII coordination with decreasing R C /R A until it again reaches the limiting situation in which all atoms mutually touch. Rotate

33. 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 R C /R A of that limiting condition?? A central cation will remain in VIII coordination with decreasing R C /R A until it again reaches the limiting situation in which all atoms mutually touch. Rotate

34. 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 R C /R A of that limiting condition?? A central cation will remain in VIII coordination with decreasing R C /R A until it again reaches the limiting situation in which all atoms mutually touch. Rotate

35. 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 R C /R A of that limiting condition?? A central cation will remain in VIII coordination with decreasing R C /R A until it again reaches the limiting situation in which all atoms mutually touch. Rotate

36. 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 R C /R A of that limiting condition?? A central cation will remain in VIII coordination with decreasing R C /R A until it again reaches the limiting situation in which all atoms mutually touch. Rotate

37. 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 R C /R A of that limiting condition?? A central cation will remain in VIII coordination with decreasing R C /R A until it again reaches the limiting situation in which all atoms mutually touch. Rotate

38. 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 R C /R A of that limiting condition?? A central cation will remain in VIII coordination with decreasing R C /R A until it again reaches the limiting situation in which all atoms mutually touch. Rotate

39. What is the R C /R A of that limiting condition?? 1.732 = d C + d A If d A = 1 then d C = 0.732 then d C = 0.732 d C /d A = R C /R A = 0.732/1 = 0.732 = 0.732/1 = 0.732 A central cation will remain in VIII coordination with decreasing R C /R A until it again reaches the limiting situation in which all atoms mutually touch. Central Plane

40. The limits for VIII coordination are thus between 1.0 (when it would by CCP or HCP) and 0.732 Note: BCC is not a cosest-packed oxygen arrangement, so it may not occur in all ionic crystal lattices

41. As R C /R A 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

46. What is the R C /R A of that limiting condition?? 1.414 = d C + d A If d A = 1 then d C = 0.414 then d C = 0.414 d C /d A = R C /R A = 0.414/1 = 0.414 = 0.414/1 = 0.414

47. As R C /R A 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

54. What is the R C /R A of the limiting condition?? Center-to-corner distance of a tetrahedron with edges of 1.0 = 0.6124 R C = 0.612 - 0.5 = 0.1124 R C /R A = 0.1124/0.5 = 0.225 = 0.1124/0.5 = 0.225

55. As R C /R A continues to decrease below the 0.22 the cation will move to the next lower coordination: III. The cation moves from the center of the tetrahedron to the center of an coplanar tetrahedral face of 3 oxygen atoms What is the R C /R A of the limiting condition?? cos 60 = 0.5/y y = 0.577 R C = 0.577 - 0.5 = 0.077 R C /R A = 0.077/0.5 = 0.155 = 0.077/0.5 = 0.155

56. If R C /R A decreases below the 0.15 (a are situation) the cation will move to the next lower coordination: II. The cation moves directly between 2 neighboring oxygen atoms

58. Homework Exercise Use R C /R Oxygen and the limits above to determine the probable coordination of the following elements in silicate and oxide minerals: Si +4 Mg 2+ Al 3+ Ti 4+ K + Ca 2+ Fe 2+ Na + Correct R C for cases in which the coordination is not VI (the standard) and recalculate the ratio