Tetragonal crystal system

Red zircon (size: 1.0 cm) from Gilgit, Pakistan

Phosgenite on display at the Royal Ontario Museum

The tetragonal crystal system is one of the seven crystal systems in crystallography . It includes all point groups that have a fourfold axis of rotation or rotation inversion in exactly one direction .

Point groups

The tetragonal crystal system includes the point groups and . They form the tetragonal crystal family and can be described with the tetragonal grid system . ${\ displaystyle \ 4, \, {\ bar {4}}, \, 4 / m, \, 422, \, 4mm, \, {\ bar {4}} 2m}$ ${\ displaystyle \ 4 / mmm}$

Grid system

The tetragonal grid system has the holoedry . Analogous to the other whirling crystal systems , the fourfold axis is placed in the direction of the c-lattice axis. As in the monoclinic, the other two directions are perpendicular to the c-axis and - due to the fourfold nature of the c-axis - must also have the same length and be perpendicular to each other. Therefore there are only the two lattice constants a and c in this crystal system and the following conditions arise: ${\ displaystyle \ 4 / mmm}$

${\ displaystyle a \ = b \ \ neq \ c \}$
${\ displaystyle \ alpha \ = \ beta \ = \ gamma \ = 90 ^ {\ circ}}$

Bravais grid

Tetragonal primitive grid: tP
Tetragonal body-centered grid: tI

In the tetragonal crystal system there are two Bravais lattices , the primitive and the body centered. The face-centered Bravais lattice does not correspond to the standard setup, since this lattice can always be described by an inner-centered lattice with half the size of the unit cell . The body-centered grid is obtained from the face-centered grid by rotating the a-axis by 45 ° around the c-axis and reducing it by the factor . ${\ displaystyle {\ frac {1} {\ sqrt {2}}}}$

Point groups in the tetragonal crystal system and their physical properties

To describe the tetragonal crystal classes in Hermann-Mauguin symbology , the symmetry operations are given with respect to given directions (viewing directions) in the grid system. The viewing direction of the first symbol is the c-axis (<001>), of the second symbol the a-axis (<100>) and of the third symbol the diagonal of the c-surface (<110>).

Characteristic for the tetragonal room groups is a 4 ( 4 ) in the first position, but not a 3 ( 3 ) in the second position of the room group symbol.

Point group (crystal class)								Physical Properties				Examples
No.	Crystal system	Surname	Schoenflies icon	International symbol ( Hermann-Mauguin )		Tepid class	Associated room groups ( no.)	Enantiomorphism	Optical activity	Pyroelectricity	Piezoelectricity ; SHG effect
No.	Crystal system	Surname	Schoenflies icon	Full	Short	Tepid class	Associated room groups ( no.)	Enantiomorphism	Optical activity	Pyroelectricity	Piezoelectricity ; SHG effect
9	tetragonal	tetragonal-pyramidal	C ₄	4th	4th	4 / m	75-80	+	+	+ [001]	+	Pinnoit Percleveit- (Ce)
10		tetragonal-disphenoidal	S ₄	4th	4th		81-82	-	+	-	+	Clerk's seat Cahnit
11		tetragonal-dipyramidal	C _{4 h}	4 / m	4 / m		83-88	-	-	-	-	Scheelite baotite
12		tetragonal-trapezoidal	D ₄	422	422	4 / mmm	89-98	+	+	-	+	Cristobalite maucherite
13		ditetragonal-pyramidal	C _{4 v}	4 mm	4 mm		99-110	-	-	+ [001]	+	Lenait Diaboleit
14th		tetragonal-scalenohedral	D _{2 d} ( V _d )	4 2 m or 4 m 2	4 2 m		111-122	-	+	-	+	Chalcopyrite stannite
15th		ditetragonal-dipyramidal	D _{4 h}	4 / m 2 / m 2 / m	4 / mmm		123-142	-	-	-	-	Rutile zircon
↑ In the information on the physical properties, " - " means forbidden due to the symmetry and " + " means allowed. No statement can be made about the magnitude of the optical activity, pyro- and piezoelectricity and the SHG effect purely due to the symmetry. However, one can assume that there is always at least a weak expression of the property. For the pyroelectricity, the direction of the pyroelectric vector is given, if available.

For further tetragonal crystallizing chemical substances see category: Tetragonal crystal system

literature

Hahn, Theo (Ed.): International Tables for Crystallography Vol. A D. Reidel publishing Company, Dordrecht 1983, ISBN 90-277-1445-2
D. Schwarzenbach: Crystallography. Springer Verlag, Berlin 2001, ISBN 3-540-67114-5
Will Kleber , Hans-Joachim Bautsch , Joachim Bohm , Detlef Klimm: Introduction to crystallography . 19th edition. Oldenbourg Wissenschaftsverlag, 2010, ISBN 978-3-486-59075-3 .
Walter Borchard-Ott: Crystallography. 7th edition. Springer Verlag, Berlin 2009, ISBN 978-3-540-78270-4

[Hinweise-1] In the information on the physical properties, " - " means forbidden due to the symmetry and " + " means allowed. No statement can be made about the magnitude of the optical activity, pyro- and piezoelectricity and the SHG effect purely due to the symmetry. However, one can assume that there is always at least a weak expression of the property. For the pyroelectricity, the direction of the pyroelectric vector is given, if available.