16 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Crystal structure | 2/5 | https://en.wikipedia.org/wiki/Crystal_structure | reference | science, encyclopedia | 2026-05-05T10:52:12.114890+00:00 | kb-cron |
Coordinates in angle brackets such as ⟨100⟩ denote a family of directions that are equivalent due to symmetry operations, such as [100], [010], [001] or the negative of any of those directions. Coordinates in curly brackets or braces such as {100} denote a family of plane normals that are equivalent due to symmetry operations, much the way angle brackets denote a family of directions. For face-centered cubic (fcc) and body-centered cubic (bcc) lattices, the primitive lattice vectors are not orthogonal. However, in these cases the Miller indices are conventionally defined relative to the lattice vectors of the cubic supercell and hence are again simply the Cartesian directions.
=== Interplanar spacing ===
The perpendicular spacing d between adjacent (hkℓ) lattice planes is given by:
Cubic:
1
d
2
=
h
2
+
k
2
+
ℓ
2
a
2
{\displaystyle {\frac {1}{d^{2}}}={\frac {h^{2}+k^{2}+\ell ^{2}}{a^{2}}}}
Tetragonal:
1
d
2
=
h
2
+
k
2
a
2
+
ℓ
2
c
2
{\displaystyle {\frac {1}{d^{2}}}={\frac {h^{2}+k^{2}}{a^{2}}}+{\frac {\ell ^{2}}{c^{2}}}}
Hexagonal:
1
d
2
=
4
3
(
h
2
+
h
k
+
k
2
a
2
)
+
ℓ
2
c
2
{\displaystyle {\frac {1}{d^{2}}}={\frac {4}{3}}\left({\frac {h^{2}+hk+k^{2}}{a^{2}}}\right)+{\frac {\ell ^{2}}{c^{2}}}}
Rhombohedral (primitive setting):
1
d
2
=
(
h
2
+
k
2
+
ℓ
2
)
sin
2
α
+
2
(
h
k
+
k
ℓ
+
h
ℓ
)
(
cos
2
α
−
cos
α
)
a
2
(
1
−
3
cos
2
α
+
2
cos
3
α
)
{\displaystyle {\frac {1}{d^{2}}}={\frac {(h^{2}+k^{2}+\ell ^{2})\sin ^{2}\alpha +2(hk+k\ell +h\ell )(\cos ^{2}\alpha -\cos \alpha )}{a^{2}(1-3\cos ^{2}\alpha +2\cos ^{3}\alpha )}}}
Orthorhombic:
1
d
2
=
h
2
a
2
+
k
2
b
2
+
ℓ
2
c
2
{\displaystyle {\frac {1}{d^{2}}}={\frac {h^{2}}{a^{2}}}+{\frac {k^{2}}{b^{2}}}+{\frac {\ell ^{2}}{c^{2}}}}
Monoclinic:
1
d
2
=
(
h
2
a
2
+
k
2
sin
2
β
b
2
+
ℓ
2
c
2
−
2
h
ℓ
cos
β
a
c
)
csc
2
β
{\displaystyle {\frac {1}{d^{2}}}=\left({\frac {h^{2}}{a^{2}}}+{\frac {k^{2}\sin ^{2}\beta }{b^{2}}}+{\frac {\ell ^{2}}{c^{2}}}-{\frac {2h\ell \cos \beta }{ac}}\right)\csc ^{2}\beta }
Triclinic:
1
d
2
=
h
2
a
2
sin
2
α
+
k
2
b
2
sin
2
β
+
ℓ
2
c
2
sin
2
γ
+
2
k
ℓ
b
c
(
cos
β
cos
γ
−
cos
α
)
+
2
h
ℓ
a
c
(
cos
γ
cos
α
−
cos
β
)
+
2
h
k
a
b
(
cos
α
cos
β
−
cos
γ
)
1
−
cos
2
α
−
cos
2
β
−
cos
2
γ
+
2
cos
α
cos
β
cos
γ
{\displaystyle {\frac {1}{d^{2}}}={\frac {{\frac {h^{2}}{a^{2}}}\sin ^{2}\alpha +{\frac {k^{2}}{b^{2}}}\sin ^{2}\beta +{\frac {\ell ^{2}}{c^{2}}}\sin ^{2}\gamma +{\frac {2k\ell }{bc}}(\cos \beta \cos \gamma -\cos \alpha )+{\frac {2h\ell }{ac}}(\cos \gamma \cos \alpha -\cos \beta )+{\frac {2hk}{ab}}(\cos \alpha \cos \beta -\cos \gamma )}{1-\cos ^{2}\alpha -\cos ^{2}\beta -\cos ^{2}\gamma +2\cos \alpha \cos \beta \cos \gamma }}}
== Classification by symmetry ==
The defining property of a crystal is its inherent symmetry. Performing certain symmetry operations on the crystal lattice leaves it unchanged. All crystals have translational symmetry in three directions, but some have other symmetry elements as well. For example, rotating the crystal 180° about a certain axis may result in an atomic configuration that is identical to the original configuration; the crystal has twofold rotational symmetry about this axis. In addition to rotational symmetry, a crystal may have symmetry in the form of mirror planes, and also the so-called compound symmetries, which are a combination of translation and rotation or mirror symmetries. A full classification of a crystal is achieved when all inherent symmetries of the crystal are identified.
=== Lattice systems === Lattice systems are a grouping of crystal structures according to the point groups of their lattice. All crystals fall into one of seven lattice systems. They are related to, but not the same as the seven crystal systems.
The most symmetric, the cubic or isometric system, has the symmetry of a cube, that is, it exhibits four threefold rotational axes oriented at 109.5° (the tetrahedral angle) with respect to each other. These threefold axes lie along the body diagonals of the cube. The other six lattice systems, are hexagonal, tetragonal, rhombohedral (often confused with the trigonal crystal system), orthorhombic, monoclinic and triclinic which is the least symmetrical as it possess only identity (E).
==== Bravais lattices ==== Bravais lattices, also referred to as space lattices, describe the geometric arrangement of the lattice points, and therefore the translational symmetry of the crystal. The three dimensions of space afford 14 distinct Bravais lattices describing the translational symmetry. All crystalline materials recognized today, not including quasicrystals, fit in one of these arrangements. The fourteen three-dimensional lattices, classified by lattice system, are shown above. The crystal structure consists of the same group of atoms, the basis, positioned around each and every lattice point. This group of atoms therefore repeats indefinitely in three dimensions according to the arrangement of one of the Bravais lattices. The characteristic rotation and mirror symmetries of the unit cell is described by its crystallographic point group.
=== Dualistic atomic ordering in ternary intermetallics === Recent developments in crystallography and intermetallic materials science have led to the identification of a new family of ternary compounds known as ZIP phases. These structures are characterised by so-called dualistic atomic ordering, in which different atomic sublattices display distinct ordering behaviour within a single crystal lattice. ZIP phases have been reported to crystallise in both face-centered cubic and hexagonal structural variants, thereby extending the known range of complex crystal architectures in intermetallic systems and illustrating the growing diversity of atomic arrangements achievable in multicomponent materials.
=== Crystal systems ===