# Lattice constant

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.

Unit cell definition using parallelepiped with lengths a, b, c and angles between the sides given by α, β, γ[1]

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 disance 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),[2] electric and magnetic fields, and its isotopic composition.[3] 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.[4][5] 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

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

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[6]

${\displaystyle V=abc.}$

## 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
Material Lattice constant (Å) Crystal structure Ref.
C (diamond) 3.567 Diamond (FCC) [7]
C (graphite) a = 2.461
c = 6.708
Hexagonal
Si 5.431020511 Diamond (FCC) [8][9]
Ge 5.658 Diamond (FCC) [8]
AlAs 5.6605 Zinc blende (FCC) [8]
AlP 5.4510 Zinc blende (FCC) [8]
AlSb 6.1355 Zinc blende (FCC) [8]
GaP 5.4505 Zinc blende (FCC) [8]
GaAs 5.653 Zinc blende (FCC) [8]
GaSb 6.0959 Zinc blende (FCC) [8]
InP 5.869 Zinc blende (FCC) [8]
InAs 6.0583 Zinc blende (FCC) [8]
InSb 6.479 Zinc blende (FCC) [8]
MgO 4.212 Halite (FCC) [10]
SiC a = 3.086
c = 10.053
Wurtzite [8]
CdS 5.8320 Zinc blende (FCC) [7]
CdSe 6.050 Zinc blende (FCC) [7]
CdTe 6.482 Zinc blende (FCC) [7]
ZnO a = 3.25
c = 5.2
Wurtzite (HCP) [11]
ZnO 4.580 Halite (FCC) [7]
ZnS 5.420 Zinc blende (FCC) [7]
PbS 5.9362 Halite (FCC) [7]
PbTe 6.4620 Halite (FCC) [7]
BN 3.6150 Zinc blende (FCC) [7]
BP 4.5380 Zinc blende (FCC) [7]
CdS a = 4.160
c = 6.756
Wurtzite [7]
ZnS a = 3.82
c = 6.26
Wurtzite [7]
AlN a = 3.112
c = 4.982
Wurtzite [8]
GaN a = 3.189
c = 5.185
Wurtzite [8]
InN a = 3.533
c = 5.693
Wurtzite [8]
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 [12]
Fe 2.856 BCC [12]
Ni 3.499 FCC [12]
Cu 3.597 FCC [12]
Mo 3.142 BCC [12]
Pd 3.859 FCC [12]
Ag 4.079 FCC [12]
W 3.155 BCC [12]
Pt 3.912 FCC [12]
Au 4.065 FCC [12]
Pb 4.920 FCC [12]
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 [13]
ZrC0.97 4.698 Halite [13]
HfC0.99 4.640 Halite [13]
VC0.97 4.166 Halite [13]
NC0.99 4.470 Halite [13]
TaC0.99 4.456 Halite [13]
Cr3C2 a = 11.47
b = 5.545
c = 2.830
Orthorhombic [13]
WC a = 2.906
c = 2.837
Hexagonal [13]
ScN 4.52 Halite [14]
LiNbO3 a = 5.1483
c = 13.8631
Hexagonal [15]
KTaO3 3.9885 Cubic perovskite [15]
BaTiO3 a = 3.994
c = 4.034
Tetragonal perovskite [15]
SrTiO3 3.98805 Cubic perovskite [15]
CaTiO3 a = 5.381
b = 5.443
c = 7.645
Orthorhombic perovskite [15]
PbTiO3 a = 3.904
c = 4.152
Tetragonal perovskite [15]
EuTiO3 7.810 Cubic perovskite [15]
SrVO3 3.838 Cubic perovskite [15]
CaVO3 3.767 Cubic perovskite [15]
BaMnO3 a = 5.673
c = 4.71
Hexagonal [15]
CaMnO3 a = 5.27
b = 5.275
c = 7.464
Orthorhombic perovskite [15]
SrRuO3 a = 5.53
b = 5.57
c = 7.85
Orthorhombic perovskite [15]
YAlO3 a = 5.179
b = 5.329
c = 7.37
Orthorhombic perovskite [15]

## References

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3. ^ 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. doi:10.1063/1.1674582
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6. ^ Dept. of Crystallography & Struc. Biol. CSIC (4 June 2015). "4. Direct and reciprocal lattices". Retrieved 9 June 2015.
7. "Lattice Constants". Argon National Labs (Advanced Photon Source). Retrieved 19 October 2014.
8. "Semiconductor NSM". Retrieved 19 October 2014.
9. ^ "Fundamental physical constants". physics.nist.gov. NIST. Retrieved 17 January 2020.
10. ^ "Substrates". Spi Supplies. Retrieved 17 May 2017.
11. ^ Hadis Morkoç and Ümit Özgur (2009). Zinc Oxide: Fundamentals, Materials and Device Technology. Weinheim: WILEY-VCH Verlag GmbH & Co.
12. 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.
13. Toth, L.E. (1967). Transition Metal Carbides and Nitrides. New York: Academic Press.
14. ^ 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.
15. 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.