Physics in a nutshell

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Symmetry, Crystal Systems and Bravais Lattices

The Role of Symmetry

During this course we will focus on discussing crystals with a discrete translational symmetry, i.e. crystals which are formed by the combination of a Bravais lattice and a corresponding basis.

Despite this restriction there are still many different lattices left satisfying the condition. However, there are some lattices types that occur particularly often in nature. Or some - actually distinct - lattices share certain properties. These aspects arise from the symmetry of the lattices.

Furthermore, when describing spatial arrangements of objects this is done by using a cordinate system. The choice of the coordinate system (e.g. cartesian, cylindrical, spherical, ...) usually depends on the symmetry as well. Therefore it is helpful to classify crystal structures according to its symmetry.

Point Symmetry

Besides the translational symmetry mentioned above we will now also make use of point symmetries, i.e. the group of symmetry operations that leaves at least one point unchanged. These comprise rotations, reflections, inversions or any combinations of these.[1] They are the basis of the classification of cystals: One can for example count the number of axes of rotation and their respective multiplicities in order to compare different crystals in regard of their symmetry.

However, I do not want to go into further details of symmetry and group theory at this point because it is rather abstract and not essential for the following. If you are interested in this you may look for a good book on crystallography.

The Seven Crystal Systems

First notice: The intention of the following listing is to give you an overview rather than making you feel required to learn them by heart!

In a first step one divides the Bravais lattices into 7 crystal systems which are defined by the lengths $a$, $b$, $c$ and angles $\alpha$, $\beta$, $\gamma$ between the primitive translation vectors. The resulting crystal systems are listed and visualised below.[2] [3] [4]

Crystal System Lengths Angles
cubic $a = b = c$ $\alpha = \beta = \gamma = 90^\circ$
trigonal $a = b = c$ $\alpha = \beta = \gamma \lt 120^\circ$, $\neq 90^\circ$
hexagonal $a = b \neq c$ $\alpha = \beta = 90^\circ$, $\gamma = 120^\circ$
tetragonal $a = b \neq c$ $\alpha = \beta = \gamma = 90^\circ$
orthorhombic $a \neq b \neq c$ $\alpha = \beta = \gamma = 90^\circ$
monoclinic $a \neq b \neq c$ $\alpha = \beta = 90^\circ \neq \gamma$
triclinic $a \neq b \neq c$ $\alpha \neq \beta \neq \gamma$
Overview over the 7 crystal systems: They are defined by the lengths and angles of the primitive translation vectors and exhibit different levels of symmetry.

The 14 Bravais Lattices

So one classifies different lattices according to the shape of the parallelepiped spanned by its primitive translation vectors.

However, this is not yet the best solution for a classification with respect to symmetry. Consider for example the unit cells (a) and (b) presented before: While cell (a) is the actual unit cell spanned by the primitive translation vectors, it does not show the symmetry of the lattice properly whereas cell (b) clearly shows the two axes of rotation.

So sometimes it makes sense not to use a primitive unit cell but one which fits better to the symmetry of the problem. This idea leads to the 14 Bravais Lattices which are depicted below ordered by the crystal systems:

Cubic

Bravais lattices in the cubic crystal system: primitive, body centered, face centered
There are three Bravais lattices with a cubic symmetry. One distinguishes the simple/primitive cubic (sc), the body centered cubic (bcc) and the face centered cubic (fcc) lattice.

Tetragonal

Bravais lattices in the tetragonal crystal system: primitive- and body centered
There are two tetragonal Bravais lattices with $a=b\neq c$ and $\alpha=\beta=\gamma=90^\circ$. One is primitive and the other body centered.

Orthorhombic

Bravais lattices in the orthorhombic crystal system: primitive, body centered, face centered and base centered
There are four orthorhombic Bravais lattices with $a\neq b\neq c$ and $\alpha=\beta=\gamma=90^\circ$: Primitive, body centered, face centered and base centered.

Hexagonal

Hexagonal Bravais lattice
When two sides are of equal length with an enclosed angle of $120^\circ$ the crystal has a hexagonal structure and thus a 6-fold rotary axis.

Monoclinic

Monoclinic Bravais lattice
As in the orthohombric structure, all edges are of unequal length. However, one of the three angles is $\neq 90^\circ$.

Trigonal and Triclinic

Trigonal and triclinic Bravais lattices
The trigonal (or rhombohedral) lattice has three edges of equal length and three equal angles ($\neq 90^\circ$). In the triclinic lattice, all edges and angles are unequal.

References

[1] R. Gross, A. Marx Festkörperphysik De Gruyter 2014 (p. 8f.)
[2] R. Gross, A. Marx Festkörperphysik De Gruyter 2014 (ch. 1.1.2.2.)
[3] S. H. Simon Oxford Solid State Basics Oxford 2013 (p. 122)
[4] S. Hunklinger Festkörperphysik De Gruyter 2014 (p. 58)

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