Unit cell/ packing efficiency

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Unit cell/ packing efficiency
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Given 8 spheres to stack, how would you do it?

• Simple cubic structure

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Coordination Polyhedra

• Consider coordination of anions about a central
  cation

Halite
Na
Cl
Cl
Cl
Cl
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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
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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 hexagonal closest packed
  structure (HCP)
• Coordination number (nearest or touching
  neighbors) 12
• 6 coplanar
• 3 above the plane
• 3 below the plane

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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)

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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)

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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)

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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

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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
• 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 cubic closest packed
  structure (CCP)
• Blue layer atoms are now in a unique position
  above voids between atoms in layers A and B

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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 cubic closest packed
  structure (CCP)
• Blue layer atoms are now in a unique position
  above voids between atoms in layers A and B

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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 cubic closest packed
  structure (CCP)
• Blue layer atoms are now in a unique position
  above voids between atoms in layers A and B

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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 cubic closest packed
  structure (CCP)
• Blue layer atoms are now in a unique position
  above voids between atoms in layers A and B

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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 cubic closest packed
  structure (CCP)
• Blue layer atoms are now in a unique position
  above voids between atoms in layers A and B

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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

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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.

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• The next smaller crystal site is

• 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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• 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
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• 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
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• 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
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• 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
35

• 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
36

• 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
37

• 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
38

• 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.

• Fe, Na will form in body centered cubic

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

• CCP coordination 12
• HCP coordination 12
• Body centered coordination 8

• Rc/Ra 1.0
• Rc/Ra 1.0
• Rc/Ra 0.732 - 1.0

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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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

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• 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

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Packing efficiency

• In 2-D
• Unstable pipes have 78. fill
• Stable pipes have 90.7 fill

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Packing efficiency

• In 3-D
• Simple cubic 52 fill
• Body-centered cubic 68 fill
• hcp and ccp 74 fill

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Common structure types

• Ccp NaCl structure
• Also called face centered cubic
• Halides, oxides, sulfides take this structure
  often

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Common structure types

• Simple cubic CsCl
• From perspective of Cs or Cl? Doesnt matter

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Common structure types

• Fluorite structure (CaF2)
• What is Ca structure?
• What type of hole does F sit in?

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Common structure types

• Fluorite structure (CaF2)
• What is Ca structure?
• What type of hole does F sit in?

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Common structure types

• Fluorite structure (CaF2)
• What is F (red) structure?
• From perspective of F, what is this structure
  like?