1. Atomic Structure

2. In ionic crystals, the ions may be considered to behave as rigid spheres. As many minerals, especially the silicate minerals, can be considered to be ionic crystals, even though this may not be
completely true; this is the ionic approximation.

3. Coordination Polyhedra
Consider coordination of anions about a central cation
Halite Na Cl Cl Cl Cl
[6]NaCl

4. 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 [6]Cl 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
Type 2
They are equivalent since you could rotate the whole structure 60o and exchange them 2 1 1

6. Closest Packing Add next layer (red) 1
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, one can only fill other dimples of that type
In this case filled all type 2 dimples
Type 1 dimples remain 1

7. Closest Packing A 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) A

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)
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 hexagonal closest packed structure (HCP)
Note top layer atoms are directly above bottom layer atoms

10. Closest Packing Unit-cell: smallest identical unit in a
Third layer:
Unit cell
Unit-cell: smallest
identical unit in a
crystal structure,
when repeated,
generates the
entire structure

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

12. A single close-packed layer

13. A single close-packed layer hexagonal unit cell

14. A single close-packed layer B positions for the next layer

15. A single close-packed layer C positions for the next layer

16. Two close-packed layers Sequence : AB

17. Two close-packed layers Sequence : AC
For 2 layers AB is the same as AC

18. Two close-packed layers Sequence : AB
Two choices for the third layer : above the A position ………...

19. Two close-packed layers Sequence : AB
Two choices for the third layer : above the A position, or above the C position

20. Hexagonal close-packing : ABABAB…...z x y

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

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 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

24. 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

25. Closest Packing A-layer C-layer B-layer A-layer
The atoms are slightly shrunken to aid in visualizing the structure
A-layer
C-layer
B-layer
A-layer

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. Cubic close packing : a smaller section of the ABC stacking

29. What happens when RC/RA decreases?
The center cation becomes too small for the [12] 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
Coordination number is now 8 (corners of cube)

31. A central cation will remain in [8]-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 [8]-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??

33. A central cation will remain in [8]-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 =

34. The limits for [8]-coordination are thus between 1
The limits for [8]-coordination are thus between 1.0 (when it would by CubicClosePacking or HexagonalClosePacking) and 0.732
Note: BodyCenteredCubic is not a closest-packed oxygen arrangement.

35. 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: [6], or octahedral. The cation is in the center of an octahedron of closest-packed oxygen atoms.
(view of face of octahedron)

36. 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: [6], or octahedral. The cation is in the center of an octahedron of closest-packed oxygen atoms.
(view of apex of octahedron)

37. Interstitial sites in close-packed structures i. e
Interstitial sites in close-packed structures i.e. the spaces between the spheres
There are two types : tetrahedral sites and octahedral sites
I : Tetrahedral sites : the small spaces between 4 close-packed spheres

38. Interstitial sites in close-packed structures
II : Octahedral sites : the larger spaces between 6 close-packed spheres

39. 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: [6], 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 =

40. 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: [4], or tetrahedral. The cation is in the center of an tetrahedron of closest-packed oxygen atoms

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: [4], 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 =

42. 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: [3]. 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 =

43. If RC/RA decreases below the 0
If RC/RA decreases below the 0.15, the cation will move to the next lower coordination: [2]. The cation moves directly between 2 neighboring oxygen atoms

44. The nickel arsenide (NiAs) structure
The structure can be described as a hexagonal close packed As ions, with Ni occupying all of the octahedral sites

45. The wurtzite (ZnS) structure
The structure can be described as a hexagonal close packed S ions, with Zn2+ ions occupying half of the tetrahedral sites

46. The sodium chloride (NaCl) structure
The structure can be described as a cubic close packed Cl- ions, with Na+ ions in all the octahedral sites

47. The sphalerite (ZnS) structure
The structure can be described as a cubic close packed S ions, with Zn2+ ions in half of the tetrahedral sites

48. Atomic and Ionic Radii
Can't absolutely determine: e- cloud is nebulous & based on probability of encountering an e-
In crystalline solids the center-to-center distance =
bond length = sum of ionic radii
How get ionic radius of X and Y in XY compound??

49. Atomic Radii a a Need one pure element
Native Cu. Atomic radius = 1/2 bond length a a

50. Ionic Radii Ionic radius of an element varies with Charge
Coordination number
Bond-type
Structure type
Effective ionic radii = empirically determined from
highly accurate structure data =
Average ionic radii for one valence and one coordination number
but includes different bond-types and different structure types

51. Ionic radii versus
coordination number

52. Pauling’s Rules for Ionic Crystals
Deal with the energy state of the crystal structure
1st Rule
The cation-anion distance =  radii
Can use RC/RA to determine the coordination number of the cation
This is our previous discussion on coordination polyhedra

53. Pauling’s Rules for Ionic Crystals
2nd Rule: The electrostatic valency principle
First note that the strength of an electrostatic bond
= valence / CN
Na+ in NaCl is in [6] coordination
For Na+ the strength =
+1 divided by 6 = + 1/6 Cl Cl Cl Cl Na

54. Pauling’s Rules for Ionic Crystals
2nd Rule: the electrostatic valency principle
An ionic structure will be stable to the extent that the sum of the strengths of electrostatic bonds that reach an anion from adjacent cations = the charge of that anion
6 ( + 1/6 ) = (sum from Na’s)
charge of Cl = -1
These charges are equal in magnitude so the structure is stable
+ 1/6 Na
+ 1/6 Na Na
Cl-
+ 1/6
+ 1/6 Na

55. Pauling’s Rules 3rd Rule:
The sharing of edges, and particularly of faces, of adjacent polyhedra tend to decrease the stability of an ionic structure

56. If the edge of the edge of the tetrahedron = 1,
B = 0.71
C = 0.58
D = base of tetrahedral poyhedron
at maximum separation of the central ion

57. Pauling’s Rules 4th Rule:
In a crystal with different cations, those of high valence and small CN tend not to share polyhedral elements
An extension of Rule 3
Si4+ in [4] coordination is very unlikely to share edges or faces

58. Pauling’s Rules 5th Rule:
The number of different kinds of constituents in a crystal tends to be small

59. Representing crystal structures as polyhedral models
In many cases it is more convenient to represent crystal structures as polyhedral models, than as ball-and-stick or space-filling models, especially when the structure is complex.
e.g. the NaCl structure can be described as an array of edge-shared octahedra
space-filling models
ball-and-stick
Polyhedral presentation

60. e.g. the sphalerite structure of ZnS can be described as an array of corner sharing tetrahedra

61. The spinel AB2O4 structure