Crystal system

 ( Symmetry ) 

In crystallography, a crystal system is a set of point groups (a group of geometric symmetries with at least one fixed point). A lattice system is a set of Bravais lattices (an infinite array of discrete points). (symmetry groups of a configuration in space) are classified into crystal systems according to their point groups, and into lattice systems according to their Bravais lattices. Crystal systems that have space groups assigned to a common lattice system are combined into a crystal family.

The seven crystal systems are triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. Informally, two crystals are in the same crystal system if they have similar symmetries (though there are many exceptions).

---

Classifications Crystals can be classified in three ways: lattice systems, crystal systems and crystal families. The various classifications are often confused: in particular the trigonal crystal system is often confused with the rhombohedral lattice system, and the term "crystal system" is sometimes used to mean "lattice system" or "crystal family".

---

Lattice system A lattice system is a group of lattices with the same set of lattice point groups. The 14 are grouped into seven lattice systems: triclinic, monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal, and cubic.

---

Crystal system A crystal system is a set of point groups in which the point groups themselves and their corresponding are assigned to a lattice system. Of the 32 crystallographic point groups that exist in three dimensions, most are assigned to only one lattice system, in which case both the crystal and lattice systems have the same name. However, five point groups are assigned to two lattice systems, rhombohedral and hexagonal, because both exhibit threefold rotational symmetry. These point groups are assigned to the trigonal crystal system.

---

Crystal family A crystal family is determined by Bravais lattice and Point group. It is formed by combining crystal systems that have space groups assigned to a common lattice system. In three dimensions, the hexagonal and trigonal crystal systems are combined into one hexagonal crystal family. crystal, with threefold c-axis symmetry]]

---

Comparison Five of the crystal systems are essentially the same as five of the lattice systems. The hexagonal and trigonal crystal systems differ from the hexagonal and rhombohedral lattice systems. These are combined into the hexagonal crystal family.

The relation between three-dimensional crystal families, crystal systems and lattice systems is shown in the following table:

Hexagonal	1 threefold axis of rotation	18
Hexagonal	Hexagonal	1 sixfold axis of rotation	7	27	1
6	7	7	Total	32	230	14

Note: there is no "trigonal" lattice system. To avoid confusion of terminology, the term "trigonal lattice" is not used.

---

Crystal classes The 7 crystal systems consist of 32 crystal classes (corresponding to the 32 crystallographic point groups) as shown in the following table below:

pinacoidal	Ci (S2)		1x	2,1+	centrosymmetric	2	Cyclic group $\backslash mathbb\{Z\}\_2$
domatic	Cs (C1h)	m	*11		polar	2	Cyclic group $\backslash mathbb\{Z\}\_2$
prismatic	C2h	2/m	2*	2,2+	centrosymmetric	4	Klein four-group $\backslash mathbb\{V\}\; =\; \backslash mathbb\{Z\}\_2\backslash times\backslash mathbb\{Z\}\_2$
rhombic-pyramidal	C2v	mm2	*22	2	polar	4	Klein four-group $\backslash mathbb\{V\}\; =\; \backslash mathbb\{Z\}\_2\backslash times\backslash mathbb\{Z\}\_2$
rhombic-	D2h (Vh)	mmm	*222	2,2	centrosymmetric	8	$\backslash mathbb\{V\}\backslash times\backslash mathbb\{Z\}\_2$
tetragonal-disphenoidal	S4		2x	2+,2	non-centrosymmetric	4	Cyclic group $\backslash mathbb\{Z\}\_4$
tetragonal-dipyramidal	C4h	4/m	4*	2,4+	centrosymmetric	8	$\backslash mathbb\{Z\}\_4\backslash times\backslash mathbb\{Z\}\_2$
tetragonal-trapezohedral	D4	422	422	2,4+	enantiomorphic	8	Dihedral group $\backslash mathbb\{D\}\_8\; =\; \backslash mathbb\{Z\}\_4\backslash rtimes\backslash mathbb\{Z\}\_2$
ditetragonal-pyramidal	C4v	4mm	*44	4	polar	8	Dihedral group $\backslash mathbb\{D\}\_8\; =\; \backslash mathbb\{Z\}\_4\backslash rtimes\backslash mathbb\{Z\}\_2$
tetragonal-scalenohedral	D2d (Vd)	2m or m2	2*2	2+,4	non-centrosymmetric	8	Dihedral group $\backslash mathbb\{D\}\_8\; =\; \backslash mathbb\{Z\}\_4\backslash rtimes\backslash mathbb\{Z\}\_2$
ditetragonal-dipyramidal	D4h	4/mmm	*422	2,4	centrosymmetric	16	$\backslash mathbb\{D\}\_8\backslash times\backslash mathbb\{Z\}\_2$
rhombohedral	C3i (S6)		3x	2+,3+	centrosymmetric	6	Cyclic group $\backslash mathbb\{Z\}\_6\; =\; \backslash mathbb\{Z\}\_3\backslash times\backslash mathbb\{Z\}\_2$
trigonal-trapezohedral	D3	32 or 321 or 312	322	3,2+	enantiomorphic	6	Dihedral group $\backslash mathbb\{D\}\_6\; =\; \backslash mathbb\{Z\}\_3\backslash rtimes\backslash mathbb\{Z\}\_2$
ditrigonal-pyramidal	C3v	3m or 3m1 or 31m	*33	3	polar	6	Dihedral group $\backslash mathbb\{D\}\_6\; =\; \backslash mathbb\{Z\}\_3\backslash rtimes\backslash mathbb\{Z\}\_2$
ditrigonal-scalenohedral	D3d	m or m1 or 1m	2*3	2+,6	centrosymmetric	12	Dihedral group $\backslash mathbb\{D\}\_\{12\}\; =\; \backslash mathbb\{Z\}\_6\backslash rtimes\backslash mathbb\{Z\}\_2$
trigonal-dipyramidal	C3h		3*	2,3+	non-centrosymmetric	6	Cyclic group $\backslash mathbb\{Z\}\_6\; =\; \backslash mathbb\{Z\}\_3\backslash times\backslash mathbb\{Z\}\_2$
hexagonal-dipyramidal	C6h	6/m	6*	2,6+	centrosymmetric	12	$\backslash mathbb\{Z\}\_6\backslash times\backslash mathbb\{Z\}\_2$
hexagonal-trapezohedral	D6	622	622	2,6+	enantiomorphic	12	Dihedral group $\backslash mathbb\{D\}\_\{12\}\; =\; \backslash mathbb\{Z\}\_6\backslash rtimes\backslash mathbb\{Z\}\_2$
dihexagonal-pyramidal	C6v	6mm	*66	6	polar	12	Dihedral group $\backslash mathbb\{D\}\_\{12\}\; =\; \backslash mathbb\{Z\}\_6\backslash rtimes\backslash mathbb\{Z\}\_2$
ditrigonal-dipyramidal	D3h	m2 or 2m	*322	2,3	non-centrosymmetric	12	Dihedral group $\backslash mathbb\{D\}\_\{12\}\; =\; \backslash mathbb\{Z\}\_6\backslash rtimes\backslash mathbb\{Z\}\_2$
dihexagonal-dipyramidal	D6h	6/mmm	*622	2,6	centrosymmetric	24	$\backslash mathbb\{D\}\_\{12\}\backslash times\backslash mathbb\{Z\}\_2$
diploidal	Th	m	3*2	3+,4	centrosymmetric	24	$\backslash mathbb\{A\}\_4\backslash times\backslash mathbb\{Z\}\_2$
gyroidal	O	432	432	4,3+	enantiomorphic	24	symmetric group $\backslash mathbb\{S\}\_4$
hextetrahedral	Td	3m	*332	3,3	non-centrosymmetric	24	symmetric group $\backslash mathbb\{S\}\_4$
hexoctahedral	Oh	mm	*432	4,3	centrosymmetric	48	$\backslash mathbb\{S\}\_4\backslash times\backslash mathbb\{Z\}\_2$

The point symmetry of a structure can be further described as follows. Consider the points that make up the structure, and reflect them all through a single point, so that ( x, y, z) becomes (− x,− y,− z). This is the 'inverted structure'. If the original structure and inverted structure are identical, then the structure is centrosymmetric. Otherwise it is non-centrosymmetric. Still, even in the non-centrosymmetric case, the inverted structure can in some cases be rotated to align with the original structure. This is a non-centrosymmetric achiral structure. If the inverted structure cannot be rotated to align with the original structure, then the structure is Chirality or enantiomorphic and its symmetry group is enantiomorphic.

A direction (meaning a line without an arrow) is called polar if its two-directional senses are geometrically or physically different. A symmetry direction of a crystal that is polar is called a polar axis. Groups containing a polar axis are called polar. A polar crystal possesses a unique polar axis (more precisely, all polar axes are parallel). Some geometrical or physical property is different at the two ends of this axis: for example, there might develop a dielectric polarization as in Pyroelectricity. A polar axis can occur only in non-centrosymmetric structures. There cannot be a mirror plane or twofold axis perpendicular to the polar axis, because they would make the two directions of the axis equivalent.

The crystal structures of chiral biological molecules (such as protein structures) can only occur in the 65 enantiomorphic space groups (biological molecules are usually chiral).

---

Bravais lattices There are seven different kinds of lattice systems, and each kind of lattice system has four different kinds of centerings (primitive, base-centered, body-centered, face-centered). However, not all of the combinations are unique; some of the combinations are equivalent while other combinations are not possible due to symmetry reasons. This reduces the number of unique lattices to the 14 Bravais lattices.

The distribution of the 14 Bravais lattices into 7 lattice systems is given in the following table.

In geometry and crystallography, a Bravais lattice is a category of translative (also known as lattices) in three directions.

Such symmetry groups consist of translations by vectors of the form

R = n₁ a₁ + n₂ a₂ + n₃ a₃,

where n₁, n₂, and n₃ are and a₁, a₂, and a₃ are three non-coplanar vectors, called primitive vectors.

These lattices are classified by the space group of the lattice itself, viewed as a collection of points; there are 14 Bravais lattices in three dimensions; each belongs to one lattice system only. They represent the maximum symmetry a structure with the given translational symmetry can have.

All crystalline materials (not including ) must, by definition, fit into one of these arrangements.

For convenience a Bravais lattice is depicted by a unit cell which is a factor 1, 2, 3, or 4 larger than the primitive cell. Depending on the symmetry of a crystal or other pattern, the fundamental domain is again smaller, up to a factor 48.

The Bravais lattices were studied by Moritz Ludwig Frankenheim in 1842, who found that there were 15 Bravais lattices. This was corrected to 14 by Auguste Bravais in 1848.

---

In other dimensions

---

Two-dimensional space In two-dimensional space, there are four crystal systems (oblique, rectangular, square, hexagonal), four crystal families (oblique, rectanguar, square, hexagonal), and four lattice systems (oblique lattice, rectangular, square lattice, and hexagonal).

(2011). 9780199573660, Oxford University Press. ISBN 9780199573660

Total	4	10	17	5

---

Four-dimensional space ‌The four-dimensional unit cell is defined by four edge lengths ( a, b, c, d) and six interaxial angles ( α, β, γ, δ, ε, ζ). The following conditions for the lattice parameters define 23 crystal families

+ Crystal families in 4D space ! No. ! Family ! Edge lengths ! Interaxial angles

The names here are given according to Whittaker.

E. J. W. Whittaker (1985). 9780198544326, Clarendon Press. ISBN 9780198544326

They are almost the same as in Brown et al.,

(1978). 9780471030959, Wiley. ISBN 9780471030959

with exception for names of the crystal families 9, 13, and 22. The names for these three families according to Brown et al. are given in parentheses.

The relation between four-dimensional crystal families, crystal systems, and lattice systems is shown in the following table. Enantiomorphic systems are marked with an asterisk. The number of enantiomorphic pairs is given in parentheses. Here the term "enantiomorphic" has a different meaning than in the table for three-dimensional crystal classes. The latter means, that enantiomorphic point groups describe chiral (enantiomorphic) structures. In the current table, "enantiomorphic" means that a group itself (considered as a geometric object) is enantiomorphic, like enantiomorphic pairs of three-dimensional space groups P3₁ and P3₂, P4₁22 and P4₃22. Starting from four-dimensional space, point groups also can be enantiomorphic in this sense.

+ Crystal systems in 4D space ! No. of crystal family ! Crystal family ! Crystal system ! Lattice system ! No. of crystal system ! Point groups ! width=120	Space groups ! Bravais lattices
Orthogonal P, S, I, Z, D, F, G, U	112	8
	Axial orthogonal		6	3	887
Hexagonal monoclinic P	15	1
	Hexagonal monoclinic		9	7	25
Tetragonal orthogonal P, S, I, Z, G	351	5
	Proper tetragonal orthogonal		13	10	1312
Hexagonal orthogonal P, S	150	2
	Hexagonal orthogonal		15	12	240
Ditetragonal orthogonal P, Z	165 (+2)	2
	Ditetragonal orthogonal		19	6	127
Dihexagonal orthogonal P	5 (+5)	1
	Dihexagonal orthogonal		23	11	20
	Ditrigonal orthogonal		22	11	41
Dihexagonal orthogonal RR		16			1
Cubic orthogonal P, I, Z, F, U	96	5
	Complex cubic orthogonal		25	11	366
Diisohexagonal orthogonal P	19 (+3)	1
	Complex diisohexagonal orthogonal		30	13 (+8)	15 (+9)
Hypercubic Z	107 (+28)	1
	Dodecagonal hypercubic		33	16 (+12)	25 (+20)
Total	23 (+6)	33 (+7)	33 (+7)		227 (+44)	4783 (+111)	64 (+10)

---

See also

---

Works cited

• (2025). 9780792365907, Springer-Verlag. . ISBN 9780792365907

---

External links

• Overview of the 32 groups
• Mineral galleries – Symmetry
• all cubic crystal classes, forms, and stereographic projections (interactive java applet)
• Crystal system at the Online Dictionary of Crystallography
• Crystal family at the Online Dictionary of Crystallography
• Lattice system at the Online Dictionary of Crystallography
• Conversion Primitive to Standard Conventional for VASP input files
• Learning Crystallography

Post Comment