Explain the seven crystal systems with neat diagrams.
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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 latticesinto 7 crystal systems which are defined by the lengths aa, bb, cc and angles αα, ββ, γγ between the primitive translation vectors. The resulting crystal systems are listed and visualised below.[2] [3] [4]
Crystal SystemLengthsAnglescubica=b=ca=b=cα=β=γ=90∘α=β=γ=90∘trigonala=b=ca=b=cα=β=γ<120∘α=β=γ<120∘, ≠90∘≠90∘hexagonala=b≠ca=b≠cα=β=90∘α=β=90∘, γ=120∘γ=120∘tetragonala=b≠ca=b≠cα=β=γ=90∘α=β=γ=90∘orthorhombica≠b≠ca≠b≠cα=β=γ=90∘α=β=γ=90∘monoclinica≠b≠ca≠b≠cα=β=90∘≠γα=β=90∘≠γtriclinica≠b≠ca≠b≠cα≠β≠γα≠β≠γOverview 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
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
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 latticesinto 7 crystal systems which are defined by the lengths aa, bb, cc and angles αα, ββ, γγ between the primitive translation vectors. The resulting crystal systems are listed and visualised below.[2] [3] [4]
Crystal SystemLengthsAnglescubica=b=ca=b=cα=β=γ=90∘α=β=γ=90∘trigonala=b=ca=b=cα=β=γ<120∘α=β=γ<120∘, ≠90∘≠90∘hexagonala=b≠ca=b≠cα=β=90∘α=β=90∘, γ=120∘γ=120∘tetragonala=b≠ca=b≠cα=β=γ=90∘α=β=γ=90∘orthorhombica≠b≠ca≠b≠cα=β=γ=90∘α=β=γ=90∘monoclinica≠b≠ca≠b≠cα=β=90∘≠γα=β=90∘≠γtriclinica≠b≠ca≠b≠cα≠β≠γα≠β≠γOverview 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
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
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