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 cationHaliteHalite

ClCl

ClCl

ClCl

ClCl

NaNa

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

Na

Na

ClCl

Coordination PolyhedraRadius 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

1122

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

11

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-layerA-layer

B-layerB-layer

C-layerC-layer

A-layerA-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 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)

• 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 = 1Set = 1

Diagonal length then = 2Diagonal length then = 2

arbitrary arbitrary since will since will deal with deal with ratiosratios

• 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 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 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 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 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 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 Rotate

• Fe, Na will form in body centered cubic

• 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.

• CCP coordination = 12

• 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 The limits for VIII coordination are thus between 1.0

(when it would by CCP or HCP) and 0.732(when it would by CCP or HCP) and 0.732

• 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 0.414 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

1y0.5

• 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 0.15 (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 holes CN = 8 or 8-fold

• Octahedral holes CN = 6 or 6-fold

• Tetrahedral holes CN = 4 or 4-fold

• CN polyhedra Rc/Ra

• 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

• 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 (CaF2)

• What is Ca structure?

• What type of hole does F sit in?

Common structure types

• Fluorite structure (CaF2)

• What is Ca structure?

• What type of hole does F sit in?

Common structure types

• Fluorite structure (CaF2)

• What is F (red) structure?

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