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

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

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

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

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)

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

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)

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)

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)

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

Closest Packing Third layer: Unit cell

Closest Packing Third layer: Unit cell

Closest Packing Third layer: Unit cell

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

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

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

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

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

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

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

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

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

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

Closest Packing Rotating toward a top view

Closest Packing Rotating toward a top view

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

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

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)

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

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

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

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

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

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

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

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

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 = 0.732

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

As RC/RA continues to decrease below the 0 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 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 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 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 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 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 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 = 0.414

As RC/RA continues to decrease below the 0 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 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 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 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 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 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 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 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 What 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 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 = 0.577 - 0.5 = 0.077 RC/RA = 0.077/0.5 = 0.155

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

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