Unit cell packing efficiency
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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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Coordination polyhedra l.jpg
Coordination Polyhedra

  • Consider coordination of anions about a central cation

Halite

Na

Cl

Cl

Cl

Cl


Coordination polyhedra4 l.jpg
Coordination Polyhedra

Na

  • 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


Slide5 l.jpg

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


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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


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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)


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Closest Packing

  • 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


Slide9 l.jpg

Closest Packing

  • Third layer:

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


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Closest Packing

  • Third layer:

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


Slide11 l.jpg

Closest Packing

  • Third layer:

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


Slide12 l.jpg

Closest Packing

  • 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


Slide13 l.jpg

Closest Packing

  • Third layer:

    • Unit cell


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Closest Packing

  • Third layer:

    • Unit cell


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Closest Packing

  • Third layer:

    • Unit cell


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Closest Packing

  • Third layer:

    • View from top shows hexagonal unit cell


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Closest Packing

  • Third layer:

    • View from top shows hexagonal unit cell

    • Mg is HCP


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Closest Packing

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


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Closest Packing

  • 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


Slide20 l.jpg

Closest Packing

  • 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


Slide21 l.jpg

Closest Packing

  • 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


Slide22 l.jpg

Closest Packing

  • 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


Slide23 l.jpg

Closest Packing

  • 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


Slide24 l.jpg

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

C-layer

B-layer

A-layer


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Closest Packing

  • Rotating toward a top view


Slide26 l.jpg

Closest Packing

  • Rotating toward a top view


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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


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Closest Packing

  • CCP is same as face centered cubic

  • Al is CCP


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  • 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.


Slide30 l.jpg

  • 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)


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  • 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


Slide32 l.jpg

  • 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


Slide33 l.jpg

  • 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


Slide34 l.jpg

  • 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


Slide35 l.jpg

  • 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


Slide36 l.jpg

  • 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


Slide37 l.jpg

  • 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


Slide38 l.jpg

  • Fe, Na will form in body centered cubic


The limits for viii coordination are thus between 1 0 when it would by ccp or hcp and 0 732 l.jpg

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


Slide40 l.jpg

  • 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


Slide41 l.jpg

  • 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


Slide42 l.jpg

  • 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


Slide43 l.jpg

  • 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


Slide44 l.jpg

  • 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


Slide45 l.jpg

  • 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


Slide46 l.jpg

  • 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


Slide47 l.jpg

  • 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


Slide48 l.jpg

  • 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


Slide49 l.jpg

  • 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


Slide50 l.jpg

  • 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


Slide51 l.jpg

  • 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


Slide52 l.jpg

  • 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


Slide53 l.jpg

  • 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


Types of coordination polyhedra voids to stuff cations into l.jpg
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


Slide55 l.jpg

  • 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 l.jpg
Packing efficiency into)

  • In 2-D

    • Unstable pipes have 78.% fill

    • Stable pipes have 90.7% fill


Packing efficiency57 l.jpg
Packing efficiency into)

  • In 3-D

    • Simple cubic 52% fill

    • Body-centered cubic 68% fill

    • hcp and ccp 74% fill


Common structure types l.jpg
Common structure types into)

  • Ccp: NaCl structure

  • Also called face centered cubic

  • Halides, oxides, sulfides take this structure often


Common structure types59 l.jpg
Common structure types into)

  • Simple cubic CsCl

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


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Common structure types into)

  • Fluorite structure (CaF2)

  • What is Ca structure?

  • What type of hole does F sit in?


Common structure types61 l.jpg
Common structure types into)

  • Fluorite structure (CaF2)

  • What is Ca structure?

  • What type of hole does F sit in?


Common structure types62 l.jpg
Common structure types into)

  • Fluorite structure (CaF2)

  • What is F (red) structure?

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


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