Consider coordination of anions about a central cationCoordination PolyhedraHaliteClClClClNa

Could do the opposite,but conventionallychoose the cationCan predict the coordination by considering the radius ratio:RC/RACations are generally smaller than anions so begin with maximum ratio = 1.0 Coordination PolyhedraNaNaNaNaCl

Coordination PolyhedraRadius Ratio: RC/RA = 1.0 (commonly native elements)Equal sized spheresClosest PackedHexagonal array:6 nearest neighbors in the planeNote dimples in which next layer atoms will settleTwo dimple types: Type 1 point NE Type 2 point SWThey are equivalent since you could rotate the whole structure 60o and exchange them12

Closest PackingAdd next layer (red)Red atoms can only settle in one dimple typeBoth types are identical and red atoms could settle in eitherOnce first red atom settles in, can only fill other dimples of that typeIn this case filled all type 2 dimples

1

Closest PackingThird layer ??Third layer dimples are now different!Call layer 1 A sitesLayer 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)

Closest PackingThird 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) = 126 coplanar3 above the plane3 below the plane

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

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

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

Closest PackingThird 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

Closest PackingThird layer:

Unit cell

Closest PackingThird layer:

Unit cell

Closest PackingThird layer:

Unit cell

Closest PackingThird layer:

View from top shows hexagonal unit cell

Closest PackingThird layer:

View from top shows hexagonal unit cell

Hexagonal Closest PackingClick to run animation Case Klein animation for Mineral Science, John Wiley & Sons

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

Closest PackingThird 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

Closest PackingThird 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

Closest PackingThird 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

Closest PackingThird 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

Closest PackingThird 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

Closest PackingView from the same side shows the face-centered cubic unit cell that results.

The atoms are slightly shrunken to aid in visualizing the structureA-layerB-layerC-layerA-layer

Closest PackingRotating toward a top view

Closest PackingRotating toward a top view

Closest PackingYou 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

Cubic Closest PackingClick to run animation Case Klein animation for Mineral Science, John Wiley & Sons

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.

Coordination PolyhedraClick to run animation Case Klein animation for Mineral Science, John Wiley & Sons

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:

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??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.Set = 1Diagonal length then = 2arbitrary since will deal with ratios

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??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.Rotate

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??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.Rotate

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??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.Rotate

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??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.Rotate

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??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.Rotate

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??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.Rotate

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??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.Rotate

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 = 0.732A 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

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

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

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

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

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

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

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 atomsWhat 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 = 0.414

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

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

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

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

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

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

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

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 atomsWhat is the RC/RA of the limiting condition??

Center-to-corner distance of a tetrahedron with edges of 1.0 = 0.6124

RC = 0.612 - 0.5 = 0.1124

RC/RA = 0.1124/0.5 = 0.225

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 atomsWhat is the RC/RA of the limiting condition??

cos 60 = 0.5/y y = 0.577

RC = 0.577 - 0.5 = 0.077

RC/RA = 0.077/0.5 = 0.155

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

Homework ExerciseUse RC/ROxygen and the limits above to determine the probable coordination of the following elements in silicate and oxide minerals:

Si+4Mg2+Al3+Ti4+K+Ca2+Fe2+Na+

Correct RC for cases in which the coordination is not VI (the standard) and recalculate the ratio