1. Anandh Subramaniam & Kantesh Balani
SUBLATTICES
&
‘SUBCRYSTALS’
MATERIALS SCIENCE
&
ENGINEERING
Anandh Subramaniam & Kantesh Balani
Materials Science and Engineering (MSE)
Indian Institute of Technology, Kanpur
URL: home.iitk.ac.in/~anandh
AN INTRODUCTORY E-BOOK
Part of
A Learner’s Guide

2. These concepts will become clear on considering examples
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.

3. X SX1 + SX2 X = SX1 + SX2 Concept of Sublattice Example-1
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 =
SX1 +
SX2
Sub-Crystal-1 (SX1) consists of only up arrows and Sub-crystal-2 (SX2) consists only of down arrows The crystal can be called a ‘Super-Crystal’ (supercrystal)

4. Correspondingly we can think of a ‘Superlattice’ (L)
Which can be broken into two Sublattices → two interpenetrating sublattices
SL1
SubLattice-1 (SL1) +
SL2
SubLattice-2 (SL2) L =
SL1 +
SL2
Sub-Lattice-1 (SL1) and Sub-Lattice-2 (SL2) combine to create the lattice (L)

5. 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

6. Example-2
Let us consider another example to understand the concept of sublattice (now in 2D)
Simple Square Crystal X
‘Super-Crystal’ (X)
This is the familiar crystal which we had considered before

7. Let us analyze this crystal in terms of subcrystals and sublattices

8. SX1
‘Super-Crystal’ (X) X =
SX1 +
SX2
SX2
Sub-Crystal-1 (SX1) consists of only green circles and Sub-crystal-2 (SX2) consists only of brown

9. SL1 L =
SL1 +
SL2
Sub-Lattice-1 (SL1) and Sub-Lattice-2 (SL2) combine to create the lattice (L)
SL2

10. L
Note that in the crystal SL2 (or equivalently SL1) is not a set of lattice points

11. 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 + =
SX1
SX2 X
‘Super-Crystal’ (X)