# Lattice, Basis and Crystal

This article is a bit technical in a sense that it provides mainly definitions of terms and concepts. This is however important in order to establish a language and avoid obscurities when talking about crystal structures.

## Lattice

A lattice is in general defined as a discrete but infinite regular arrangement of points (lattice sites) in a vector space^{[1]}

### Bravais Lattice

In solid state physics one usually encounters lattices which exhibit a discrete translational symmetry. If one considers for instance the vector space $\mathbb{R}^3$ this means that a translation of the whole lattice by any translation vector given by \begin{align} \vec{T}_{mno} = m \, \vec{a}_1 + n \, \vec{a}_2 + o \,\vec{a}_3 \qquad m,n,o \in \mathbb{Z} \; \label{eq:threeDimTranslVector} \end{align} leaves the lattice unchanged.

A lattice that can be characterised in this way is referred to as a Bravais lattice.^{[2]} All lattice points are equivalent, i.e. all properties remain invariant under translations by any vector $\vec{T}_{mno}$. Also, an observer sitting on one specific lattice point would see the same environment as when sitting on any other. However, if there are lattice points with different environments they cannot form a Bravais lattice! (This is an equivalent definition of a Bravais lattice.^{[3]} ^{[4]} ^{[5]} )

### Primitive Translation Vectors

The linear independent generating vectors $\vec{a}_i$ of the translations \eqref{eq:threeDimTranslVector} are called primitive translation vectors or primitive lattice vectors.^{[6]} ^{[7]} However, for a given lattice the choice of the primitive translation vectors is not unique!^{[8]} ^{[9]}

Let's illustrate this for the two-dimensional Euclidean space: Any two linear independent vectors $\vec{a}_1$ and $\vec{a}_2$ form a basis and can be used as generators for a two-dimensional lattice:

The lattice can then be described as a linear combination \begin{align} \vec{T}_{mn} = m \, \vec{a}_1 + n \, \vec{a}_2 \qquad m,n \in \mathbb{Z} \label{eq:twoDimTranslVector} \end{align} of these vectors with $m$ and $n$ being integer coefficients. The same applies for the $\mathbb{R}^3$ when adding a third linear independent vector $\vec{a}_3$ as in eq. \eqref{eq:threeDimTranslVector}.

### Definition of a Lattice in the Literature

If you go through literature on solid state physics you may notice that the terms lattice and Bravais lattice are sometimes used interchangeably.

The lattice definition according to eq. \eqref{eq:threeDimTranslVector} is indeed more correct from a mathematical point of view and even more common among crystallographers. However, objects such as a honeycomb would then not be regarded as a lattice. But this is just a matter of taste, so don't let yourself be confused by this!^{[10]}

## Basis and Crystal

Now one could go ahead and replace the lattice points by more complex objects (called basis), e.g. a group of atoms, a molecule, ... . This generates a structure that is referred to as a crystal:^{[11]} ^{[12]} ^{[13]} ^{[14]}

For many solids it is a good approximation to look at them as crystals since they consist of small groups of atoms that are arranged in a repetitive manner.

## References

[1] | Festkörperphysik De Gruyter 2014 (ch. 1.1) |

[2] | Quantum Chemistry of Solids Springer 2012 (ch. 1.2.1) |

[3] | Festkörperphysik De Gruyter 2014 (ch. 1.1.1) |

[4] | The Physics of Solids Springer 2010 (ch. 3.3.1) |

[5] | Solid-State Physics Wiley 2006 (ch. 1.2.1) |

[6] | Einführung in die Festkörperphysik Oldenbourg 2006 (p- 4) |

[7] | Festkörperphysik De Gruyter 2014 (ch. 1.1.1) |

[8] | Oxford Solid State Basics Oxford 2013 (p. 113) |

[9] | The Physics of Solids Springer 2010 (ch. 3.3.1) |

[10] | Oxford Solid State Basics Oxford 2013 (p. 113) |

[11] | Festkörperphysik De Gruyter 2014 (ch. 1.1) |

[12] | Oxford Solid State Basics Oxford 2013 (p. 116) |

[13] | The Physics of Solids Springer 2010 (ch 3.3.1) |

[14] | Solid-State Physics Wiley 2006 (ch. 1.2.2) |