A lattice constant or lattice parameter is one of the physical dimensions and angles that determine the geometry of the unit cells in a crystal lattice, and is proportional to the distance between atoms in the crystal. A simple cubic crystal has only one lattice constant, the distance between atoms, but, in general, lattices in three dimensions have six lattice constants: the lengths a, b, and c of the three cell edges meeting at a vertex, and the angles α, β, and γ between those edges.

The crystal lattice parameters a, b, and c have the dimension of length. The three numbers represent the size of the unit cell, that is, the distance from a given atom to an identical atom in the same position and orientation in a neighboring cell (except for very simple crystal structures, this will not necessarily be distance to the nearest neighbor). Their SI unit is the meter, and they are traditionally specified in angstroms (Å); an angstrom being 0.1 nanometer (nm), or 100 picometres (pm). Typical values start at a few angstroms. The angles α, β, and γ are usually specified in degrees.

Introduction A chemical substance in the solid state may form crystals in which the atoms, molecules, or ions are arranged in space according to one of a small finite number of possible crystal systems (lattice types), each with fairly well defined set of lattice parameters that are characteristic of the substance. These parameters typically depend on the temperature, pressure (or, more generally, the local state of mechanical stress within the crystal),Francisco Colmenero (2019): "Negative area compressibility in oxalic acid dihydrate". Materials Letters, volume 245, pages 25-28. electric and magnetic fields, and its isotopic composition.Roland Tellgren and Ivar Olovsson (1971): "Hydrogen Bond Studies. XXXXVI. The Crystal Structures of Normal and Deuterated Sodium Hydrogen Oxalate Monohydrate NaHC2O4·H2O and NaDC2O4·D2O". Journal of Chemical Physics, volume 54, issue 1. The lattice is usually distorted near impurities, crystal defects, and the crystal's surface. Parameter values quoted in manuals should specify those environment variables, and are usually averages affected by measurement errors.

Depending on the crystal system, some or all of the lengths may be equal, and some of the angles may have fixed values. In those systems, only some of the six parameters need to be specified. For example, in the cubic system, all of the lengths are equal and all the angles are 90°, so only the a length needs to be given. This is the case of diamond, which has at 300 K. Similarly, in hexagonal system, the a and b constants are equal, and the angles are 60°, 90°, and 90°, so the geometry is determined by the a and c constants alone.

The lattice parameters of a crystalline substance can be determined using techniques such as X-ray diffraction or with an atomic force microscope. They can be used as a natural length standard of nanometer range. In the epitaxial growth of a crystal layer over a substrate of different composition, the lattice parameters must be matched in order to reduce strain and crystal defects.

Volume The volume of the unit cell can be calculated from the lattice constant lengths and angles. If the unit cell sides are represented as vectors, then the volume is the scalar triple product of the vectors. The volume is represented by the letter V. For the general unit cell

For monoclinic lattices with , , this simplifies to

For orthorhombic, tetragonal and cubic lattices with as well, then

Lattice matching

Matching of lattice structures between two different semiconductor materials allows a region of band gap change to be formed in a material without introducing a change in crystal structure. This allows construction of advanced light-emitting diodes and diode lasers.

For example, gallium arsenide, aluminium gallium arsenide, and aluminium arsenide have almost equal lattice constants, marking it possible to grow almost arbitrarily thick layers of one on the other one.

Lattice grading Typically, films of different materials grown on the previous film or substrate are chosen to match the lattice constant of the prior layer to minimize film stress.

An alternative method is to grade the lattice constant from one value to another by a controlled altering of the alloy ratio during film growth. The beginning of the grading layer will have a ratio to match the underlying lattice and the alloy at the end of the layer growth will match the desired final lattice for the following layer to be deposited.

The rate of change in the alloy must be determined by weighing the penalty of layer strain, and hence defect density, against the cost of the time in the epitaxy tool.

For example, indium gallium phosphide layers with a band gap above 1.9 eV can be grown on gallium arsenide wafers with index grading.

List of lattice constants +Lattice constants for various materials at 300 K Material Lattice constant (Å) Crystal structure Ref. C (diamond) 3.567 Diamond (FCC) C (graphite) a = 2.461c = 6.708 Hexagonal Si 5.431020511 Diamond (FCC) Ge 5.658 Diamond (FCC) AlAs 5.6605 Zinc blende (FCC) AlP 5.4510 Zinc blende (FCC) AlSb 6.1355 Zinc blende (FCC) GaP 5.4505 Zinc blende (FCC) GaAs 5.653 Zinc blende (FCC) GaSb 6.0959 Zinc blende (FCC) InP 5.869 Zinc blende (FCC) InAs 6.0583 Zinc blende (FCC) InSb 6.479 Zinc blende (FCC) MgO 4.212 Halite (FCC) SiC a = 3.086c = 10.053 Wurtzite CdS 5.8320 Zinc blende (FCC) CdSe 6.050 Zinc blende (FCC) CdTe 6.482 Zinc blende (FCC) ZnO a = 3.25c = 5.2 Wurtzite (HCP) ZnO 4.580 Halite (FCC) ZnS 5.420 Zinc blende (FCC) PbS 5.9362 Halite (FCC) PbTe 6.4620 Halite (FCC) BN 3.6150 Zinc blende (FCC) BP 4.5380 Zinc blende (FCC) CdS a = 4.160c = 6.756 Wurtzite ZnS a = 3.82c = 6.26 Wurtzite AlN a = 3.112c = 4.982 Wurtzite GaN a = 3.189c = 5.185 Wurtzite InN a = 3.533c = 5.693 Wurtzite LiF4.03 Halite LiCl5.14 Halite LiBr5.50 Halite LiI6.01 Halite NaF4.63 Halite NaCl5.64 Halite NaBr5.97 Halite NaI6.47 Halite KF5.34 Halite KCl6.29 Halite KBr6.60 Halite KI7.07 Halite RbF5.65 Halite RbCl6.59 Halite RbBr6.89 Halite RbI7.35 Halite CsF6.02 Halite CsCl4.123 Caesium chloride CsBr4.291 Caesium chloride CsI4.567 Caesium chloride Al4.046FCC Fe2.856BCC Ni3.499FCC Cu3.597FCC Mo3.142BCC Pd3.859FCC Ag4.079FCC W3.155BCC Pt3.912FCC Au4.065FCC Pb4.920FCC V3.0399BCC Nb3.3008BCC Ta3.3058BCC TiN4.249Halite ZrN4.577Halite HfN4.392Halite VN4.136Halite CrN4.149Halite NbN4.392Halite TiC4.328Halite 4.698Halite 4.640Halite 4.166Halite 4.470Halite 4.456Halite a = 11.47b = 5.545c = 2.830Orthorhombic WCa = 2.906c = 2.837Hexagonal ScN4.52Halite a = 5.1483c = 13.8631Hexagonal 3.9885Cubic perovskite a = 3.994c = 4.034Tetragonal perovskite 3.98805Cubic perovskite a = 5.381b = 5.443c = 7.645Orthorhombic perovskite a = 3.904c = 4.152Tetragonal perovskite 7.810Cubic perovskite 3.838Cubic perovskite 3.767Cubic perovskite a = 5.673c = 4.71Hexagonal a = 5.27b = 5.275c = 7.464Orthorhombic perovskite a = 5.53b = 5.57c = 7.85Orthorhombic perovskite a = 5.179b = 5.329c = 7.37Orthorhombic perovskite

References

External links How to Find Lattice Constant