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

2. Coordination Polyhedra
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

3. 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

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 A-layer C-layer B-layer A-layer
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

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 RC/RA 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. The next smaller crystal site is:
Body-Centered Cubic (BCC) with cation (red) in the center of a cube
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.
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

39. 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.
Central Plane
What is the RC/RA of that limiting condition??
1.732 = dC + dA
If dA = 1
then dC = 0.732
dC/dA = RC/RA
= 0.732/1 =

40. The limits for VIII coordination are thus between 1
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 RC/RA continues to decrease below the 0
As RC/RA continues to decrease below the 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
As RC/RA continues to decrease below the 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
As RC/RA continues to decrease below the 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
As RC/RA continues to decrease below the 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
As RC/RA continues to decrease below the 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. As RC/RA continues to decrease below the 0
As RC/RA continues to decrease below the 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
What is the RC/RA of that limiting condition??
1.414 = dC + dA
If dA = 1
then dC = 0.414
dC/dA = RC/RA
= 0.414/1 =

47. As RC/RA continues to decrease below the 0
As RC/RA continues to decrease below the 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
As RC/RA continues to decrease below the 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
As RC/RA continues to decrease below the 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
As RC/RA continues to decrease below the 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

51. As RC/RA continues to decrease below the 0
As RC/RA continues to decrease below the 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

52. As RC/RA continues to decrease below the 0
As RC/RA continues to decrease below the 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

53. As RC/RA continues to decrease below the 0
As RC/RA continues to decrease below the 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. As RC/RA continues to decrease below the 0
As RC/RA continues to decrease below the 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
What is the RC/RA of the limiting condition??
Center-to-corner distance of a tetrahedron with edges of 1.0 =
RC = =
RC/RA
= /0.5 =

55. As RC/RA continues to decrease below the 0
As RC/RA 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 RC/RA of the limiting condition??
cos 60 = 0.5/y y = 0.577
RC = = 0.077
RC/RA
= 0.077/0.5 =

56. If RC/RA decreases below the 0
If RC/RA 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 Si+4 Mg2+ Al3+ Ti4+ K+ Ca2+ Fe2+ Na+
Use RC/ROxygen and the limits above to determine the probable coordination of the following elements in silicate and oxide minerals:
Si+4 Mg2+
Al3+ Ti4+
K+ Ca2+
Fe2+ Na+
Correct RC for cases in which the coordination is not VI (the standard) and recalculate the ratio