# Lattices

(Redirected from Lattice)
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)
• ${\displaystyle \left\langle hkl\right\rangle }$ 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
• ${\displaystyle \left\langle hkl\right\rangle }$ 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:

## Peak Positions

### Cubic

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

### Hexagonal

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

### Tetragonal

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

### Orthorhombic

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