SUBLATTICES

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SUBLATTICES SUBCRYSTALS
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Sublattices and Subcrystals

• The concept of sublattices (and a new concept of
  subcrystals based on this) are useful in
  understanding ordered structures.
• The use of the term superlattice implies that it
  is composed of more than one sublattice.
• Typically all sublattices are identical, but with
  the origin of one shifted w.r.t to the other.
• Populating a sublattice with a species/motif (a
  sub-motif?) gives us a subcrystal.
• Subcrystals may be identical (same species sits
  in both the subcrystals) or may be different
  (sib-motif populating the sublattices may be
  different).
• Subcrystals combine (interpenetrate) to give a
  supercrystal (analogous to the superlattice)

Click here to see connection between
superlattices and ordered structures
These concepts will become clear on considering
examples
Usually the use of the prefix super implies
an highly enhanced property, like in
superconductivity, superfluidity,
superparamagnetism etc. In the case of the
superlattice it just implies that it is made of
more than one sublattice Sub-motif may be
thought of as a part of the motif of the
supercrystal.
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Example-1
Concept of Sublattice
Let us revisit the crystal (X) made of up arrows
and down arrows to understand the concept of
sublattices
X
Super-Crystal (X)
This crystal can be understood as a superposition
of two crystals as below
SX1
Sub-Crystal-1 (SX1)

SX2
Sub-Crystal-2 (SX2)
X

SX2
SX1

Sub-Crystal-1 (SX1) consists of only up arrows
and Sub-crystal-2 (SX2) consists only of down
arrowsThe crystal can be called a
Super-Crystal (supercrystal)
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Correspondingly we can think of a Superlattice
(L)
L
Lattice
Which can be broken into two Sublattices ? two
interpenetrating sublattices
SL1
SubLattice-1 (SL1)

SL2
SubLattice-2 (SL2)
L

SL2
SL1

Sub-Lattice-1 (SL1) and Sub-Lattice-2 (SL2)
combine to create the lattice (L)
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If the lattice parameter of the crystal is a
then Sublattice-1 (SL1) is displaced with respect
to Sublattice-2 (SL2) by a/2
Note that in the crystal SL2 (or equivalently
SL1) is not a set of lattice points
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Example-2
Let us consider another example to understand the
concept of sublattice (now in 2D)
Square Crystal
X
Super-Crystal (X)
This is the familiar crystal which we had
considered before
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Let us analyze this crystal in terms of
subcrystals and sublattices
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SX1
Super-Crystal (X)
X

SX2
SX1

SX2
Sub-Crystal-1 (SX1) consists of only green
circles and Sub-crystal-2 (SX2) consists only of
brown
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SL1
L

SL2
SL1

Sub-Lattice-1 (SL1) and Sub-Lattice-2 (SL2)
combine to create the lattice (L)
SL2
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L
Note that in the crystal SL2 (or equivalently
SL1) is not a set of lattice points
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Example-3
Let us consider a 3D example of a Supercrystal
(superlattice)
This crystal can be thought of a two
interpenetrating subcrystals SX1 FCC SL1
decorated by white metallic balls SX2 FCC SL2
decorated by brown metallic balls
NaCl
If the brown spheres are Na ions and white
spheres are Cl? ions (of different sizes) this
can be thought of as a model for NaCl


X
SX2
SX1
Super-Crystal (X)