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-copolymer mesophases can be thought of as nanostructures sitting on lattice sites (e.g. cylinders in a hexagonal lattice).

Well-defined 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

Example of an alternating simple cubic (NaCl) lattice.

Cubic

There are three cubic space groups:

• Simple cubic (SC)
  • Alternating simple cubic (NaCl)
• Body-centered cubic (BCC)
  • CsCl
• Face-centered cubic (FCC)

There are many conceptually distinct lattices that exhibit one of the above cubic symmetries:

• Diamond lattice (FCC)
  • B32 (NaTl)

Hexagonal

• Hexagonal
  • Hexagonal close-packed (HCP)
  • Hexagonal diamond
    • Wurtzite
  • AlB₂

Symmetry and Space Groups

• U. Shmueli, S. R. Hall and R. W. Grosse-Kunstleve, Symmetry in reciprocal space International Tables for Crystallography Volume B: Reciprocal space, 1, Springer 2001 pp 99-161 ISSN 1574-8707 ISBN 978-0-7923-6592-1 (Print) 978-1-4020-5407-5 (Online) doi 10.1107/97809553602060000552

Peak Positions

Cubic

${\frac {1}{d_{hkl}^{2}}}={\frac {h^{2}+k^{2}+l^{2}}{a^{2}}}$

$q_{hkl}=2\pi \left({\frac {h^{2}+k^{2}+l^{2}}{a^{2}}}\right)^{1/2}$

Hexagonal

${\frac {1}{d_{hkl}^{2}}}={\frac {4}{3}}\left({\frac {h^{2}+hk+k^{2}}{a^{2}}}\right)+{\frac {l^{2}}{c^{2}}}$

$q_{hkl}=2\pi \left({\frac {4(h^{2}+hk+k^{2})}{3a^{2}}}+{\frac {l^{2}}{c^{2}}}\right)^{1/2}$

Tetragonal

${\frac {1}{d_{hkl}^{2}}}={\frac {h^{2}+k^{2}}{a^{2}}}+{\frac {l^{2}}{c^{2}}}$

$q_{hkl}=2\pi \left({\frac {h^{2}+k^{2}}{a^{2}}}+{\frac {l^{2}}{c^{2}}}\right)^{1/2}$

Orthorhombic

${\frac {1}{d_{hkl}^{2}}}={\frac {h^{2}}{a^{2}}}+{\frac {k^{2}}{b^{2}}}+{\frac {l^{2}}{c^{2}}}$

$q_{hkl}^{2}=2\pi \left({\frac {h^{2}}{a^{2}}}+{\frac {k^{2}}{b^{2}}}+{\frac {l^{2}}{c^{2}}}\right)^{1/2}$

See Also

• Wikipedia: Crystal Structure
• Unit cell
• Lattice:Packing fraction
• Lattices of nano-objects
• Space Group List
• M. Nespolo, M. I. Aroyo and B. Souvignier Crystallographic shelves: space-group hierarchy explained J. Appl. Cryst. 2018 doi: 10.1107/S1600576718012724
• M. Nespolo Lattice versus structure, dimensionality versus periodicity: a crystallographic Babel? J. Appl. Cryst. 2019, 52. doi: 10.1107/S1600576719000463