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## Summary

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. Lattices in three dimensions generally 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. Unit cell definition using parallelepiped with lengths a, b, c and angles between the sides given by α, β, γ

The crystal lattice parameters a, b, and c have the dimension of length. 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.

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), electric and magnetic fields, and its isotopic composition. 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 a = 3.57 Å = 357 pm 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

$V=abc{\sqrt {1+2\cos \alpha \cos \beta \cos \gamma -\cos ^{2}\alpha -\cos ^{2}\beta -\cos ^{2}\gamma }}.$

For monoclinic lattices with α = 90°, γ = 90°, this simplifies to

$V=abc\sin \beta .$

For orthorhombic, tetragonal and cubic lattices with β = 90° as well, then

$V=abc.$

## 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, making it possible to grow almost arbitrarily thick layers of one on the other one.

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 = 2.461
c = 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 = 3.086
c = 10.053
Wurtzite 
CdS 5.8320 Zinc blende (FCC) 
CdSe 6.050 Zinc blende (FCC) 
CdTe 6.482 Zinc blende (FCC) 
ZnO a = 3.25
c = 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 = 4.160
c = 6.756
Wurtzite 
ZnS a = 3.82
c = 6.26
Wurtzite 
AlN a = 3.112
c = 4.982
Wurtzite 
GaN a = 3.189
c = 5.185
Wurtzite 
InN a = 3.533
c = 5.693
Wurtzite 
LiF 4.03 Halite
LiCl 5.14 Halite
LiBr 5.50 Halite
LiI 6.01 Halite
NaF 4.63 Halite
NaCl 5.64 Halite
NaBr 5.97 Halite
NaI 6.47 Halite
KF 5.34 Halite
KCl 6.29 Halite
KBr 6.60 Halite
KI 7.07 Halite
RbF 5.65 Halite
RbCl 6.59 Halite
RbBr 6.89 Halite
RbI 7.35 Halite
CsF 6.02 Halite
CsCl 4.123 Caesium chloride
CsI 4.567 Caesium chloride
Al 4.046 FCC 
Fe 2.856 BCC 
Ni 3.499 FCC 
Cu 3.597 FCC 
Mo 3.142 BCC 
Pd 3.859 FCC 
Ag 4.079 FCC 
W 3.155 BCC 
Pt 3.912 FCC 
Au 4.065 FCC 
Pb 4.920 FCC 
V 3.0399 BCC
Nb 3.3008 BCC
Ta 3.3058 BCC
TiN 4.249 Halite
ZrN 4.577 Halite
HfN 4.392 Halite
VN 4.136 Halite
CrN 4.149 Halite
NbN 4.392 Halite
TiC 4.328 Halite 
ZrC0.97 4.698 Halite 
HfC0.99 4.640 Halite 
VC0.97 4.166 Halite 
NC0.99 4.470 Halite 
TaC0.99 4.456 Halite 
Cr3C2 a = 11.47
b = 5.545
c = 2.830
Orthorombic 
WC a = 2.906
c = 2.837
Hexagonal 
ScN 4.52 Halite 
LiNbO3 a = 5.1483
c = 13.8631
Hexagonal 
KTaO3 3.9885 Cubic perovskite 
BaTiO3 a = 3.994
c = 4.034
Tetragonal perovskite 
SrTiO3 3.98805 Cubic perovskite 
CaTiO3 a = 5.381
b = 5.443
c = 7.645
Orthorhombic perovskite 
PbTiO3 a = 3.904
c = 4.152
Tetragonal perovskite 
EuTiO3 7.810 Cubic perovskite 
SrVO3 3.838 Cubic perovskite 
CaVO3 3.767 Cubic perovskite 
BaMnO3 a = 5.673
c = 4.71
Hexagonal 
CaMnO3 a = 5.27
b = 5.275
c = 7.464
Orthorhombic perovskite 
SrRuO3 a = 5.53
b = 5.57
c = 7.85
Orthorhombic perovskite 
YAlO3 a = 5.179
b = 5.329
c = 7.37
Orthorhombic perovskite