# 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:

## 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

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