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 Cl Cl Cl Cl Na

Coordination Polyhedra Could do the opposite, but conventionally choose the cation Can predict the coordination by considering the radius ratio: R C /R A Cations are generally smaller than anions so begin with maximum ratio = 1.0 Na Cl

Coordination Polyhedra Radius Ratio: R C /R A = 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 60 o and exchange them 1 2

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 –Mg is HCP

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 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 B-layer C-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

Closest Packing CCP is same as face centered cubic Al is CCP

What happens when R C /R A 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) 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 R C /R A of that limiting condition?? A central cation will remain in VIII coordination with decreasing R C /R A until it again reaches the limiting situation in which all atoms mutually touch. Set = 1 Diagonal length then = 2 arbitrary 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 R C /R A of that limiting condition?? A central cation will remain in VIII coordination with decreasing R C /R A 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 R C /R A of that limiting condition?? A central cation will remain in VIII coordination with decreasing R C /R A 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 R C /R A of that limiting condition?? A central cation will remain in VIII coordination with decreasing R C /R A 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 R C /R A of that limiting condition?? A central cation will remain in VIII coordination with decreasing R C /R A 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 R C /R A of that limiting condition?? A central cation will remain in VIII coordination with decreasing R C /R A 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 R C /R A of that limiting condition?? A central cation will remain in VIII coordination with decreasing R C /R A until it again reaches the limiting situation in which all atoms mutually touch. Rotate

Fe, Na will form in body centered cubic A central cation will remain in VIII coordination with decreasing R C /R A until it again reaches the limiting situation in which all atoms mutually touch.

CCP coordination = 12 HCP coordination = 12 Body centered coordination = 8 Rc/Ra = 1.0 Rc/Ra = The limits for VIII coordination are thus between 1.0 (when it would by CCP or HCP) and 0.732

As R C /R A 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

As R C /R A 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

As R C /R A 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 R C /R A of the limiting condition?? cos 60 = 0.5/y y = R C = = R C /R A = 0.077/0.5 = 0.155

If R C /R A 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

Types of coordination polyhedra (voids to stuff cations into) Cubic holesCN = 8 or 8-fold Octahedral holesCN = 6 or 6-fold Tetrahedral holes CN = 4 or 4-fold

CNpolyhedraRc/Ra 3triangular tetrahedral octahedral cubic HCP or CCP1.0

Packing efficiency In 2-D –Unstable pipes have 78.% fill –Stable pipes have 90.7% fill

Packing efficiency In 3-D –Simple cubic 52% fill –Body-centered cubic 68% fill –hcp and ccp 74% fill

Common structure types Ccp: NaCl structure Also called face centered cubic Halides, oxides, sulfides take this structure often

Common structure types Simple cubic CsCl From perspective of Cs or Cl? Doesn’t matter

Common structure types Fluorite structure (CaF 2 ) What is Ca structure? What type of hole does F sit in?

Common structure types Fluorite structure (CaF 2 ) What is Ca structure? What type of hole does F sit in?

Common structure types Fluorite structure (CaF 2 ) What is F (red) structure? From perspective of F, what is this structure like?