Close Packed Crystals

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Close Packed Crystals
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Close Packed Crystals

• Initially we consider usual type of close
  packed crystals, which are made of single kind of
  sphere.
• In other types of close packed crystals (e.g.
  tetrahedrally close packed crystals, also called
  topologically close packed crystals), more than
  one size of sphere may be involved.
• One may even conceive of close packing of
  ellipsoids and other non-spherical objects.

• Cubic Close Packed (CCP- commonly called FCC
  crystal also) and Hexagonal Close Packed (HCP)
  are two common examples of close packed crystals.
• The term close packed crystal implies closest
  packed crystal (having a packing fraction of
  0.74).
• The proof that this is the densest
  crystallographic packing of spheres possible is a
  difficult one (and will not be considered here).
• CCP and HCP are just two examples among a series
  of close packed structures which can be envisaged
  (shown in coming slides).
• Every atom in these structures has a coordination
  number of 12 ? forming a Cubeoctahedron or a
  Twinned Cubeoctahedron (around the central atom).

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• The common starting point is a close packed layer
  of atoms with 6-fold symmetry.
• Identical layers are stacked one on another with
  a shift.
• The shift is such that the atoms in the above
  (and below) layers sit in the valleys formed by
  a layer.
• All such possibilities (see coming slides) lead
  to Close Packed Crystals.
• The original 6-fold symmetry present in a single
  layer is lost on this kind of packing (you must
  be aware of the 3-fold present in CCP and HCP
  crystals!). Yes! HCP crystal has NO true 6-fold
  axis!

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? Coordination Polyhedron ? Cubeoctahedron
CCP
HCP
? Coordination Polyhedron ? Twinned Cubeoctahedron
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• Starting Point ? Hexagonal layer
• Three positions A (the first layer atomic
  positions), B C (Valleys) are shown
• The second layer (of hexagonal packing of atoms)
  can be positioned in valley B (or equivalently in
  valley C)

Part of the hexagonal layer shown
Step-1
A
Step-2
AB
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• The third layer can be positioned with atoms
  directly above the A layer (Option-1) or with
  atoms above the C layer (Option-2)

Layer-3
ABA
(Option-1)
Continuing this ABAB sequence we get the HCP
structure
C-site vacant
Step-3
(Option-2)
ABC
Continuing this ABCABC sequence we get the CCP
structure (Though not obvious!)
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• ABCABC ABAB are just but two amongst the
  infinite possibilities
• At each stage of construction we have a choice of
  putting an atomic layer at A, B or C position
• Possibilites include? ABCAB.ABCAB.ABCAB ?
  ABCABCAB.ABCABCAB.ABCABCAB
• Hence we can construct crystals with larger and
  larger unit cells.
• If we randomly put the layers we will not get a
  crystal in the true sense.(We can think of
  these as 2D crystals, which are not periodic in
  3rd dimension).
• Few stages in the infinite choice tree is shown
  below.

B
A
Track a branch to infinity or truncate at some
stage and repeat to get a structure
C
B
A
C
B
A
B
A
C
C
B
B
C
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• In the ABCABC packing we start with a layer
  having 6-fold symmetry. Interestingly, this
  packing leads to a 4-fold axis at an angle of
  54.74? to the original 6-fold axis and to the
  familiar Cubic Close Packed crystal (FCC unit
  cell)

Actually a 3 (3 bar) roto-inversion pseudo-axis
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• Rigid sphere-like atoms without long range
  interactions can arrange in any of the infinite
  possibilities shown before.
• Not only can we have ordered sequences, but also
  disordered close packed sequences (the diorder is
  in the way A, B C appear and not within a
  given plane (say A)
• If Cobalt is annealed above and below 450?C a
  disordered sequence of ABC packing is obtained (T
  gt450?C Co ? ABCABC packing, T lt450?C Co ? ABAB
  packing)

Some examples of various stacking sequences
Layers Stacking Example Stacking symbol
2 AB Mg (hP2, P63/mmc) 2H
3 ABC Cu (cF4, Fm3m) 3C
4 ABAC La (hp4, P63/mmc) 4H
9 ABABCBCAC Sm (hR3, R3m) 9R

• Lipson Stokes (Proc. Roy. Soc. A, 181,
  101. 1943) showed the formation of trigonal
  graphite with stacking sequence ABCA instead of
  ABAB.Note Graphite is not a close packed
  structure.
• SiC (not close packed structure) shows many
  polytypes. Common ones are 3C-SiC (cubic unit
  cell, zincblende) 2H-SiC 4H-SiC 6H-SiC
  (hexagonal unit cell, wurtzile ) 15R-SiC
  (rhombohedral unit cell).
• Among the polytypes of diamond the following is
  the decreasing order of stability 3C gt 6H gt 9R gt
  4H gt 2H.

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La
Closed packed crystal
0001
C layer
A layer
B layer
A layer

Lattice parameter(s) a 3.77Å, c 12.159Å
Space Group P63/mmc (194)
Strukturbericht notation
Pearson symbol hp4
Other examples with this structure
Wyckoff position SiteSymmetry x y z Occupancy
La1 2a -3m 0 0 0 1
La2 2c -6m2 0.33 0.67 0.25 1
Note All atoms are La
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More views
C layer
B
A layer
C
B layer
A
A layer