Lattices

Example of a BCC lattice.

In x-ray scattering, we frequently study materials which have constituents arranged on a well-defined lattice. For instance, an atomic crystal has atoms which occupy well-defined sites within a representative unit cell, which then repeats in all three directions throughout space. Nanoparticle superlattices are a nanoscale analogue, where each lattice site is occupied by a nanoparticle. Other kinds of nanostructures systems can be considered similarly. Block-copolymers mesophases can be thought of as nanostructures sitting on lattice sites (e.g. cylinders in a hexagonal lattice).

Well-define realspace lattices (repeating structures) give rise to well-defined peaks in reciprocal-space, which makes it possible to determine the realspace lattice by considering the arrangement (symmetry) of the scattering peaks.

Notation

• Real space:
  • Crystal planes:
    • (hkl) denotes a plane of the crystal structure (and repetitions of that plane, with the given spacing). In cubic systems (but not others), the normal to the plane is [hkl]
    • {hkl} denotes the set of all planes that are equivalent to (hkl) by the symmetry of the lattice
  • Crystal directions:
    • [hkl] denotes a direction of a vector (in the basis of the direct lattice vectors)
    • $\left\langle hkl\right\rangle$ denotes the set of all directions that are equivalent to [hkl] by symmetry (e.g. in cubic system 〈100〉 means [100], [010], [001], [-100], [0-10], [00-1])
  • hkl denotes a diffracting plane

• Reciprocal space:
  • Reciprocal planes:
    • [hkl] denotes a plane
    • $\left\langle hkl\right\rangle$ denotes the set of all planes that are equivalent to [hkl]
  • Reciprocal directions:
    • (hkl) denotes a particular direction (normal to plane (hkl) in real space)
    • {hkl} denotes the set of all directions that are equivalent to (hkl)
  • hkl denotes an indexed reflection

Lattices

Cubic

There are three cubic space groups:

• Simple cubic (SC)
• Body-centered cubic (BCC)
• Face-centered cubic (FCC)

There are other lattices that have cubic symmetry:

• Diamond lattice (FCC)

Hexagonal

TBD

• Hexagonal diamond

See Also

• Wikipedia: Crystal Structure
• Lattice:Packing fraction