# Lattice constant

The lattice constant, or lattice parameter, refers to the physical dimension of unit cells in a crystal lattice. Lattices in three dimensions generally have three lattice constants, referred to as a, b, and c. However, in the special case of cubic crystal structures, all of the constants are equal and are referred to as a. Similarly, in hexagonal crystal structures, the a and b constants are equal, and we only refer to the a and c constants. A group of lattice constants could be referred to as lattice parameters. However, the full set of lattice parameters consist of the three lattice constants and the three angles between them.

For example, The lattice constant for diamond is a = 3.57 Å at 300 K. The structure is equilateral although its actual shape cannot be determined from only the lattice constant. Furthermore, in real applications, typically the average lattice constant is given. Near the crystal's surface, lattice constant is affected by the surface reconstruction that results in a deviation from its mean value. This deviation is especially important in nanocrystals since surface-to-nanocrystal core ratio is large.[2] As lattice constants have the dimension of length, their SI unit is the meter. Lattice constants are typically on the order of several ångströms (i.e. tenths of a nanometer). Lattice constants can be determined using techniques such as X-ray diffraction or with an atomic force microscope. Lattice constant of a crystal can be used as a natural length standard of nanometer range.[3][4]

In epitaxial growth, the lattice constant is a measure of the structural compatibility between different materials. Lattice constant matching is important for the growth of thin layers of materials on other materials; when the constants differ, strains are introduced into the layer, which prevents epitaxial growth of thicker layers without 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 dot product of one vector with the cross product of the other two vectors. The volume is represented by the letter V. For the general unit cell

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

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

${\displaystyle V=abc\sin \beta .}$

For orthorhombic, tetragonal and cubic lattices with β = 90° as well, then[5]

${\displaystyle V=abc.}$

## 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
MaterialLattice constant (Å)Crystal structureRef.
C (diamond)3.567Diamond (FCC)[6]
C (graphite)a = 2.461
c = 6.708
Hexagonal
Si5.431Diamond (FCC)[7]
Ge5.658Diamond (FCC)[7]
AlAs5.6605Zinc blende (FCC)[7]
AlP5.4510Zinc blende (FCC)[7]
AlSb6.1355Zinc blende (FCC)[7]
GaP5.4505Zinc blende (FCC)[7]
GaAs5.653Zinc blende (FCC)[7]
GaSb6.0959Zinc blende (FCC)[7]
InP5.869Zinc blende (FCC)[7]
InAs6.0583Zinc blende (FCC)[7]
InSb6.479Zinc blende (FCC)[7]
MgO4.212Halite (FCC)[8]
SiCa = 3.086
c = 10.053
Wurtzite[7]
CdS5.8320Zinc blende (FCC)[6]
CdSe6.050Zinc blende (FCC)[6]
CdTe6.482Zinc blende (FCC)[6]
ZnOa = 3.25
c = 5.2
Wurtzite (HCP)[9]
ZnO4.580Halite (FCC)[6]
ZnS5.420Zinc blende (FCC)[6]
PbS5.9362Halite (FCC)[6]
PbTe6.4620Halite (FCC)[6]
BN3.6150Zinc blende (FCC)[6]
BP4.5380Zinc blende (FCC)[6]
CdSa = 4.160
c = 6.756
Wurtzite[6]
ZnSa = 3.82
c = 6.26
Wurtzite[6]
AlNa = 3.112
c = 4.982
Wurtzite[7]
GaNa = 3.189
c = 5.185
Wurtzite[7]
InNa = 3.533
c = 5.693
Wurtzite[7]
LiF4.03Halite
LiCl5.14Halite
LiBr5.50Halite
LiI6.01Halite
NaF4.63Halite
NaCl5.64Halite
NaBr5.97Halite
NaI6.47Halite
KF5.34Halite
KCl6.29Halite
KBr6.60Halite
KI7.07Halite
RbF5.65Halite
RbCl6.59Halite
RbBr6.89Halite
RbI7.35Halite
CsF6.02Halite
CsCl4.123Caesium chloride
CsI4.567Caesium chloride
Al4.046FCC[10]
Fe2.856BCC[10]
Ni3.499FCC[10]
Cu3.597FCC[10]
Mo3.142BCC[10]
Pd3.859FCC[10]
Ag4.079FCC[10]
W3.155BCC[10]
Pt3.912FCC[10]
Au4.065FCC[10]
Pb4.920FCC[10]
TiN4.249Halite
ZrN4.577Halite
HfN4.392Halite
VN4.136Halite
CrN4.149Halite
NbN4.392Halite
TiC4.328Halite[11]
ZrC0.974.698Halite[11]
HfC0.994.640Halite[11]
VC0.974.166Halite[11]
NC0.994.470Halite[11]
TaC0.994.456Halite[11]
Cr3C2a = 11.47
b = 5.545
c = 2.830
Orthorombic[11]
WCa = 2.906
c = 2.837
Hexagonal[11]
ScN4.52Halite[12]
LiNbO3a = 5.1483
c = 13.8631
Hexagonal[13]
KTaO33.9885Cubic perovskite[13]
BaTiO3a = 3.994
c = 4.034
Tetragonal perovskite[13]
SrTiO33.98805Cubic perovskite[13]
CaTiO3a = 5.381
b = 5.443
c = 7.645
Orthorhombic perovskite[13]
PbTiO3a = 3.904
c = 4.152
Tetragonal perovskite[13]
EuTiO37.810Cubic perovskite[13]
SrVO33.838Cubic perovskite[13]
CaVO33.767Cubic perovskite[13]
BaMnO3a = 5.673
c = 4.71
Hexagonal[13]
CaMnO3a = 5.27
b = 5.275
c = 7.464
Orthorhombic perovskite[13]
SrRuO3a = 5.53
b = 5.57
c = 7.85
Orthorhombic perovskite[13]
YAlO3a = 5.179
b = 5.329
c = 7.37
Orthorhombic perovskite[13]

## References

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2. Abdulsattar, Mudar A. (2011). "Ab initio large unit cell calculations of the electronic structure of diamond nanocrystals". Solid State Sci. 13 (5): 843. Bibcode:2011SSSci..13..843A. doi:10.1016/j.solidstatesciences.2011.03.009.
3. R. V. Lapshin (1998). "Automatic lateral calibration of tunneling microscope scanners" (PDF). Review of Scientific Instruments. USA: AIP. 69 (9): 3268–3276. Bibcode:1998RScI...69.3268L. doi:10.1063/1.1149091. ISSN 0034-6748.
4. R. V. Lapshin (2019). "Drift-insensitive distributed calibration of probe microscope scanner in nanometer range: Real mode". Applied Surface Science. Netherlands: Elsevier B. V. 470: 1122–1129. arXiv:1501.06679. doi:10.1016/j.apsusc.2018.10.149. ISSN 0169-4332.
5. Dept. of Crystallography & Struc. Biol. CSIC (4 June 2015). "4. Direct and reciprocal lattices". Retrieved 9 June 2015.
6. "Lattice Constants". Argon National Labs (Advanced Photon Source). Retrieved 19 October 2014.
7. "Semiconductor NSM". Retrieved 19 October 2014.
8. "Substrates". Spi Supplies. Retrieved 17 May 2017.
9. Hadis Morkoç and Ümit Özgur (2009). Zinc Oxide: Fundamentals, Materials and Device Technology. Weinheim: WILEY-VCH Verlag GmbH & Co.
10. Davey, Wheeler (1925). "Precision Measurements of the Lattice Constants of Twelve Common Metals". Physical Review. 25 (6): 753–761. Bibcode:1925PhRv...25..753D. doi:10.1103/PhysRev.25.753.
11. Toth, L.E. (1967). Transition Metal Carbides and Nitrides. New York: Academic Press.
12. Saha, B. (2010). "Electronic structure, phonons, and thermal properties of ScN, ZrN, and HfN: A first-principles study" (PDF). Journal of Applied Physics. 107 (3): 033715–033715–8. Bibcode:2010JAP...107c3715S. doi:10.1063/1.3291117.
13. Goodenough, J. B.; Longo, M. "3.1.7 Data: Crystallographic properties of compounds with perovskite or perovskite-related structure, Table 2 Part 1". SpringerMaterials - The Landolt-Börnstein Database.