Difference between revisions of "Lattices"
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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|Block-copolymer]] mesophases can be thought of as nanostructures sitting on lattice sites (e.g. cylinders in a hexagonal 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|Block-copolymer]] mesophases can be thought of as nanostructures sitting on lattice sites (e.g. cylinders in a hexagonal lattice). | ||
| − | Well-define realspace lattices (repeating structures) [[Fourier transform|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. | + | Well-define [[realspace]] lattices (repeating structures) [[Fourier transform|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== | ==Notation== | ||
Revision as of 11:27, 30 September 2014
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-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.
Contents
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)
- 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
- Crystal planes:
- Reciprocal space:
- Reciprocal planes:
- [hkl] denotes a plane
- 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
- Reciprocal planes:
Lattices
Cubic
There are three cubic space groups:
There are many conceptually distinct lattices that exhibit one of the above cubic symmetries:
- Diamond lattice (FCC)
