Neumann's principle

The Neumann principle is a symmetry principle . It links the structure of a crystal with its physical properties.

Neumann's principle states that the symmetry of the physical properties of a crystal must contain the symmetry elements of the point group of the crystal.

Franz Ernst Neumann formulated this principle in the context of his lectures at the University of Königsberg in 1873/74. It was published in print in 1885. The final version comes from the famous textbook on crystal physics by his student Woldemar Voigt . This refers to an article by Neumann from 1833 in which Neumann had already implicitly applied this principle.

Pierre Curie extended this principle to the Curie principle in 1894 . By representation theory , these considerations are made on an advanced mathematical basis.

Explanation

The physical properties of a crystal are generally anisotropic. They depend on the direction of the acting force as well as on the direction of the examined effect. Therefore these properties are described with the help of tensors . Neumann's principle requires that every symmetry mapping of the crystal must also be a symmetry mapping of this tensor. These symmetry considerations lead to the fact that the number of independent elements of a tensor is reduced in more highly symmetrical crystals and that its main axes lie in the direction of the crystal axes. Examples:

• Since the directional dependence of the growth rate of the crystal is subject to Neumann's principle, the crystal faces reflect the point group of the crystal. So one can already deduce the crystal system and the crystal class from the crystal shape.
• A consequence of Neumann's principle that is often mentioned is the fact that no piezoelectric effect can exist in a crystal that has a center of inversion.

Neumann's principle only determines the minimum symmetry of the tensor. The tensor can, however, have additional symmetries. The deformation tensor is basically centrosymmetric due to its definition.

example

One consequence of Neumann's principle is that in the axis system given by the conventional unit cell , the property tensors have a shape that is determined by the respective crystal class. In the following tables in the monoclinic crystal system the monoclinic axis is placed in the crystallographic c-axis (1st setting).

1st order tensors

The pyroelectric effect is described by a polar vector . Accordingly, the pyromagnetic effect would have to be described by an axial vector . So far, however, there are no studies available. The point groups with a non-vanishing element of a polar vector are also called polar point groups.

General form of vectors and pseudo-vectors

Crystal system	Crystal class	Components of a vector	Number of independent components	Components of a pseudo-vector	Number of independent components
Triclinic	${\displaystyle 1\ }$	${\displaystyle {\begin{pmatrix}v_{1}\\v_{2}\\v_{3}\\\end{pmatrix}}}$	3	${\displaystyle {\begin{pmatrix}p_{1}\\p_{2}\\p_{3}\\\end{pmatrix}}}$	3
	${\displaystyle {\bar {1}}}$	${\displaystyle {\begin{pmatrix}0\\0\\0\\\end{pmatrix}}}$	0	${\displaystyle {\begin{pmatrix}p_{1}\\p_{2}\\p_{3}\\\end{pmatrix}}}$	3
Monoclinic	${\displaystyle 2\ }$	${\displaystyle {\begin{pmatrix}0\\0\\v_{3}\\\end{pmatrix}}}$	1	${\displaystyle {\begin{pmatrix}0\\0\\p_{3}\\\end{pmatrix}}}$	1
	${\displaystyle m\ }$	${\displaystyle {\begin{pmatrix}v_{1}\\v_{2}\\0\\\end{pmatrix}}}$	2	${\displaystyle {\begin{pmatrix}0\\0\\p_{3}\\\end{pmatrix}}}$	1
	${\displaystyle 2/m\ }$	${\displaystyle {\begin{pmatrix}0\\0\\0\\\end{pmatrix}}}$	0	${\displaystyle {\begin{pmatrix}0\\0\\p_{3}\\\end{pmatrix}}}$	1
Orthorhombic	${\displaystyle 222\ }$	${\displaystyle {\begin{pmatrix}0\\0\\0\\\end{pmatrix}}}$	0	${\displaystyle {\begin{pmatrix}0\\0\\0\\\end{pmatrix}}}$	0
	${\displaystyle mm2\ }$	${\displaystyle {\begin{pmatrix}0\\0\\v_{3}\\\end{pmatrix}}}$	1	${\displaystyle {\begin{pmatrix}0\\0\\0\\\end{pmatrix}}}$	0
	${\displaystyle m\ m\ m}$	${\displaystyle {\begin{pmatrix}0\\0\\0\\\end{pmatrix}}}$	0	${\displaystyle {\begin{pmatrix}0\\0\\0\\\end{pmatrix}}}$	0
Tetragonal	${\displaystyle 4\ }$	${\displaystyle {\begin{pmatrix}0\\0\\v_{3}\\\end{pmatrix}}}$	1	${\displaystyle {\begin{pmatrix}0\\0\\p_{3}\\\end{pmatrix}}}$	1
	${\displaystyle 422\ }$	${\displaystyle {\begin{pmatrix}0\\0\\v_{3}\\\end{pmatrix}}}$	1	${\displaystyle {\begin{pmatrix}0\\0\\0\\\end{pmatrix}}}$	0
	${\displaystyle 4mm\ }$	${\displaystyle {\begin{pmatrix}0\\0\\v_{3}\\\end{pmatrix}}}$	1	${\displaystyle {\begin{pmatrix}0\\0\\0\\\end{pmatrix}}}$	0
	${\displaystyle 4/m\ }$	${\displaystyle {\begin{pmatrix}0\\0\\0\\\end{pmatrix}}}$	0	${\displaystyle {\begin{pmatrix}0\\0\\p_{3}\\\end{pmatrix}}}$	1
	${\displaystyle {\bar {4}}}$	${\displaystyle {\begin{pmatrix}0\\0\\0\\\end{pmatrix}}}$	0	${\displaystyle {\begin{pmatrix}0\\0\\p_{3}\\\end{pmatrix}}}$	1
	${\displaystyle {\bar {4}}m2\ }$	${\displaystyle {\begin{pmatrix}0\\0\\0\\\end{pmatrix}}}$	0	${\displaystyle {\begin{pmatrix}0\\0\\0\\\end{pmatrix}}}$	0
	${\displaystyle 4/m\ m\ m}$	${\displaystyle {\begin{pmatrix}0\\0\\0\\\end{pmatrix}}}$	0	${\displaystyle {\begin{pmatrix}0\\0\\0\\\end{pmatrix}}}$	0
Trigonal	${\displaystyle 3\!}$	${\displaystyle {\begin{pmatrix}0\\0\\v_{3}\\\end{pmatrix}}}$	1	${\displaystyle {\begin{pmatrix}0\\0\\p_{3}\\\end{pmatrix}}}$	1
	${\displaystyle 32\ }$	${\displaystyle {\begin{pmatrix}0\\0\\0\\\end{pmatrix}}}$	0	${\displaystyle {\begin{pmatrix}0\\0\\0\\\end{pmatrix}}}$	0
	${\displaystyle 3m\ }$	${\displaystyle {\begin{pmatrix}0\\0\\v_{3}\\\end{pmatrix}}}$	1	${\displaystyle {\begin{pmatrix}0\\0\\0\\\end{pmatrix}}}$	0
	${\displaystyle {\bar {3}}}$	${\displaystyle {\begin{pmatrix}0\\0\\0\\\end{pmatrix}}}$	0	${\displaystyle {\begin{pmatrix}0\\0\\p_{3}\\\end{pmatrix}}}$	1
	${\displaystyle {\bar {3}}m}$	${\displaystyle {\begin{pmatrix}0\\0\\0\\\end{pmatrix}}}$	0	${\displaystyle {\begin{pmatrix}0\\0\\0\\\end{pmatrix}}}$	0
Hexagonal	${\displaystyle 6\ }$	${\displaystyle {\begin{pmatrix}0\\0\\v_{3}\\\end{pmatrix}}}$	1	${\displaystyle {\begin{pmatrix}0\\0\\p_{3}\\\end{pmatrix}}}$	1
	${\displaystyle 622\ }$	${\displaystyle {\begin{pmatrix}0\\0\\0\\\end{pmatrix}}}$	0	${\displaystyle {\begin{pmatrix}0\\0\\0\\\end{pmatrix}}}$	0
	${\displaystyle 6mm\ }$	${\displaystyle {\begin{pmatrix}0\\0\\v_{3}\\\end{pmatrix}}}$	1	${\displaystyle {\begin{pmatrix}0\\0\\0\\\end{pmatrix}}}$	0
	${\displaystyle 6/m\ }$	${\displaystyle {\begin{pmatrix}0\\0\\0\\\end{pmatrix}}}$	0	${\displaystyle {\begin{pmatrix}0\\0\\p_{3}\\\end{pmatrix}}}$	1
	${\displaystyle {\bar {6}}}$	${\displaystyle {\begin{pmatrix}0\\0\\0\\\end{pmatrix}}}$	0	${\displaystyle {\begin{pmatrix}0\\0\\p_{3}\\\end{pmatrix}}}$	1
	${\displaystyle {\bar {6}}m2}$	${\displaystyle {\begin{pmatrix}0\\0\\0\\\end{pmatrix}}}$	0	${\displaystyle {\begin{pmatrix}0\\0\\0\\\end{pmatrix}}}$	0
	${\displaystyle 6/m\ m\ m\ }$	${\displaystyle {\begin{pmatrix}0\\0\\0\\\end{pmatrix}}}$	0	${\displaystyle {\begin{pmatrix}0\\0\\0\\\end{pmatrix}}}$	0
Cubic	${\displaystyle 23\ }$	${\displaystyle {\begin{pmatrix}0\\0\\0\\\end{pmatrix}}}$	0	${\displaystyle {\begin{pmatrix}0\\0\\0\\\end{pmatrix}}}$	0
	${\displaystyle m{\bar {3}}}$	${\displaystyle {\begin{pmatrix}0\\0\\0\\\end{pmatrix}}}$	0	${\displaystyle {\begin{pmatrix}0\\0\\0\\\end{pmatrix}}}$	0
	${\displaystyle 432\ }$	${\displaystyle {\begin{pmatrix}0\\0\\0\\\end{pmatrix}}}$	0	${\displaystyle {\begin{pmatrix}0\\0\\0\\\end{pmatrix}}}$	0
	${\displaystyle {\bar {4}}3m}$	${\displaystyle {\begin{pmatrix}0\\0\\0\\\end{pmatrix}}}$	0	${\displaystyle {\begin{pmatrix}0\\0\\0\\\end{pmatrix}}}$	0
	${\displaystyle m{\bar {3}}m}$	${\displaystyle {\begin{pmatrix}0\\0\\0\\\end{pmatrix}}}$	0	${\displaystyle {\begin{pmatrix}0\\0\\0\\\end{pmatrix}}}$	0

2nd order tensors

The thermal expansion and the dielectric constant are described by a symmetrical tensor 2nd level. Without further restrictions this tensor has 6 independent components and an arbitrary position to the crystal axes. In the individual crystal systems, it takes the following form based on Neumann's principle:

Crystal system	Tensor surface	Component scheme	Relation to the main values	Number of independent components
Triclinic	Triaxial ellipsoid in any position	${\displaystyle {\begin{pmatrix}\epsilon _{11}&\epsilon _{12}&\epsilon _{13}\\\epsilon _{12}&\epsilon _{22}&\epsilon _{23}\\\epsilon _{13}&\epsilon _{23}&\epsilon _{33}\end{pmatrix}}}$	-	6th
Monoclinic	Triaxial ellipsoid, one major axis parallel to the b-axis	${\displaystyle {\begin{pmatrix}\epsilon _{11}&0&\epsilon _{13}\\0&\epsilon _{22}&0\\\epsilon _{13}&0&\epsilon _{33}\end{pmatrix}}}$	${\displaystyle \epsilon _{22}=\epsilon _{b}}$	4th
Orthorhombic	Triaxial ellipsoid, major axes parallel to the grid axes	${\displaystyle {\begin{pmatrix}\epsilon _{11}&0&0\\0&\epsilon _{22}&0\\0&0&\epsilon _{33}\end{pmatrix}}}$	${\displaystyle \epsilon _{11}=\epsilon _{a}}$ ${\displaystyle \epsilon _{22}=\epsilon _{b}}$ ${\displaystyle \epsilon _{33}=\epsilon _{c}}$	3
Tetragonal Trigonal Hexagonal	Ellipsoid of revolution. Axis of rotation parallel to c	${\displaystyle {\begin{pmatrix}\epsilon _{11}&0&0\\0&\epsilon _{11}&0\\0&0&\epsilon _{33}\end{pmatrix}}}$	${\displaystyle \epsilon _{11}=\epsilon _{a}=\epsilon _{b}}$ ${\displaystyle \epsilon _{33}=\epsilon _{c}}$	2
Cubic	Bullet	${\displaystyle {\begin{pmatrix}\epsilon _{11}&0&0\\0&\epsilon _{11}&0\\0&0&\epsilon _{11}\end{pmatrix}}}$	${\displaystyle \epsilon _{11}=\epsilon }$	1

Individual evidence

1. ↑ Neumann FE, 1885, lectures on the theory of the elasticity of solid bodies and light ether , OE Meyer. Leipzig (ed.), BG Teubner-Verlag.

literature

• Will Kleber , Hans-Joachim Bautsch , Joachim Bohm , Detlef Klimm: Introduction to crystallography . 19th edition. Oldenbourg Wissenschaftsverlag, Munich 2010, ISBN 978-3-486-59075-3 .
• Woldemar Voigt: Textbook of crystal physics. Leipzig 1910.

Web links

• Neumann's principle