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Unit cell/ packing efficiency. Given 8 spheres to stack, how would you do it?. Simple cubic structure. Coordination Polyhedra. Consider coordination of anions about a central cation. Halite. Na. Cl. Cl. Cl. Cl. Coordination Polyhedra. Na. Could do the opposite, but conventionally

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### Unit cell/ packing efficiency

• Simple cubic structure

• Consider coordination of anions about a central cation

Halite

Na

Cl

Cl

Cl

Cl

Na

• Could do the opposite,

but conventionally

choose the cation

• Can predict the coordination

RC/RA

Cations are generally smaller than anions so begin with maximum ratio = 1.0

Na

Na

Cl

Na

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

• 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

• 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)

• Third layer:

• If occupy A-type site the layer ordering becomes A-B-A-B and creates a hexagonalclosest packedstructure(HCP)

• Coordination number (nearest or touching neighbors) = 12

• 6 coplanar

• 3 above the plane

• 3 below the plane

• Third layer:

• If occupy A-type site the layer ordering becomes A-B-A-B and creates a hexagonalclosest packedstructure(HCP)

• Third layer:

• If occupy A-type site the layer ordering becomes A-B-A-B and creates a hexagonalclosest packedstructure(HCP)

• Third layer:

• If occupy A-type site the layer ordering becomes A-B-A-B and creates a hexagonalclosest packedstructure(HCP)

• Third layer:

• If occupy A-type site the layer ordering becomes A-B-A-B and creates a hexagonalclosest packedstructure(HCP)

• Note top layer atoms are directly above bottom layer atoms

• Third layer:

• Unit cell

• Third layer:

• Unit cell

• Third layer:

• Unit cell

• Third layer:

• View from top shows hexagonal unit cell

• Third layer:

• View from top shows hexagonal unit cell

• Mg is HCP

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

• Third layer:

• If occupy C-type site the layer ordering is A-B-C-A-B-C and creates a cubicclosest packedstructure(CCP)

• Blue layer atoms are now in a unique position above voids between atoms in layers A and B

• Third layer:

• If occupy C-type site the layer ordering is A-B-C-A-B-C and creates a cubicclosest packedstructure(CCP)

• Blue layer atoms are now in a unique position above voids between atoms in layers A and B

• Third layer:

• If occupy C-type site the layer ordering is A-B-C-A-B-C and creates a cubicclosest packedstructure(CCP)

• Blue layer atoms are now in a unique position above voids between atoms in layers A and B

• Third layer:

• If occupy C-type site the layer ordering is A-B-C-A-B-C and creates a cubicclosest packedstructure(CCP)

• Blue layer atoms are now in a unique position above voids between atoms in layers A and B

• Third layer:

• If occupy C-type site the layer ordering is A-B-C-A-B-C and creates a cubicclosest packedstructure(CCP)

• Blue layer atoms are now in a unique position above voids between atoms in layers A and B

• 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

• Rotating toward a top view

• Rotating toward a top view

• 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

• CCP is same as face centered cubic

• Al is CCP

• What happens when RC/RA decreases?

• The center cation becomes too small for the site (as if a hard-sphere atom model began to rattle in the 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.

• Body-Centered Cubic (BCC) with cation (red) in the center of a cube

• All cations need to be the same element for BCC

• Coordination number is now 8 (corners of cube)

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• Fe, Na will form in body centered cubic

CCP coordination = 12 decreasing R

HCP coordination = 12

Body centered coordination = 8

Rc/Ra = 1.0

Rc/Ra = 1.0

Rc/Ra = 0.732 - 1.0

The limits for VIII coordination are thus between 1.0 (when it would by CCP or HCP) and 0.732

• As R decreasing 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 R decreasing 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 R decreasing 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 R decreasing 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 R decreasing 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 R decreasing 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 R decreasing 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 R decreasing 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 R decreasing 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 R decreasing 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 R decreasing 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 R decreasing 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 R decreasing 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 R decreasing 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

• Cubic holes CN = 8 or 8-fold

• Octahedral holes CN = 6 or 6-fold

• Tetrahedral holes CN = 4 or 4-fold

• CN polyhedra Rc/Ra into)

• 3 triangular 0.155-0.225

• 4 tetrahedral 0.225-0.414

• 6 octahedral 0.414-0.732

• 8 cubic 0.732-1.0

• 12 HCP or CCP 1.0

Packing efficiency into)

• In 2-D

• Unstable pipes have 78.% fill

• Stable pipes have 90.7% fill

Packing efficiency into)

• In 3-D

• Simple cubic 52% fill

• Body-centered cubic 68% fill

• hcp and ccp 74% fill

• Ccp: NaCl structure

• Also called face centered cubic

• Halides, oxides, sulfides take this structure often

• Simple cubic CsCl

• From perspective of Cs or Cl? Doesn’t matter

• Fluorite structure (CaF2)

• What is Ca structure?

• What type of hole does F sit in?

• Fluorite structure (CaF2)

• What is Ca structure?

• What type of hole does F sit in?

• Fluorite structure (CaF2)

• What is F (red) structure?

• From perspective of F, what is this structure like?