Lattices

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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:

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

Hexagonal

Symmetry and Space Groups

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