Category: Advanced Chemistry

Tetragonal unit cells are found in space lattices that are formed by stacking the same two-dimensional lattices as those for space lattices containing orthorhombic unit cells, i.e. either figure II or figure III.

For space lattices containing orthorhombic unit cells, the layers of either figure II or figure III are stacked such that IaI ≠ IcI. If the layers are stacked at a height such that IaI = IcI, we get two new unit cells: primitive tetragonal (IIf) and body-centred tetragonal (IIg), both of which have the parameters IaI ≠ IbI = IcI and α = β = γ = 90^o.

The unit cell IIh that is derived from the stacking of either IIe or IIIa (with IaI = IcI), is equivalent to the primitive tetragonal unit cell IIf. Similarly, the unit cell IIIi that is derived from the stacking of IIIc (with IaI = IcI) is equivalent to the body-centred tetragonal unit cell IIg.

We can rename the basis vectors so that the parameters for a primitive tetragonal unit cell and and those for a body-centred tetragonal unit cell are IaI = IbI ≠ IcI and α = β = γ = 90^o, which are the conventional parameters of a tetragonal unit cell. Both tetragonal unit cells can also be formed by stacking layers of the two-dimensional lattice of figure IV (see diagram below).

Finally, a tetragonal unit cell has one 4-fold rotational axis of symmetry since it has two opposite faces that are squares.

 

next article: hexagonal and rhombohedral

Previous article: Orthorhombic

Content page of X-ray crystallography

Content page of advanced chemistry

Main content page

Hexagonal and rhombohedral unit cells are fundamental structures in crystallography that exemplify the diversity of atomic arrangements in solid materials.

The simplest way to form a three-dimensional Bravais lattice with 6-fold rotational symmetry is to stack the lattices of figure V one above another (Va), giving the primitive hexagonal unit cell (demarcated by red lines in Vb) with IaI = IbI ≠ IcI and α = β = 90^o, γ = 120^o.

A three-dimensional Bravais lattice with 3-fold rotational symmetry is formed by the staggered stacking of layers of figure V, with equal spacing between layers. The lateral position of the second layer is such that the lattice points of this layer are above the middle of equilateral triangles formed by the first layer (Vc). The third layer is also staggered in the similar way with its lattice points above the middle of equilateral triangles formed by the first as well as the second layer. This is known as the triple hexagonal lattice.

If we continue this manner of staggered-stacking, we have the lattice points of fourth layer lying directly above those of the first layer, giving an XYZXYZ stacking arrangement and producing the primitive rhombohedral unit cell (Vd) with IaI = IbI = IcI and α = β = γ ≠ 90^o.

Note that the triple hexagonal lattice (Vc) does not have any 6-fold rotational symmetry, since a rotation of 60⁰ using an axis perpendicular to the plane of the page and through the red lattice point maps lattice point 1 of the second layer to lattice point 2 of the third layer and not to lattice point 3.

 

next article: Cubic

Previous article: tetragonal

Content page of X-ray crystallography

Content page of advanced chemistry

Main content page

There are 3 types of cubic unit cells: primitive, body-centred and face-centred.

A three-dimensional Bravais lattice with 4-fold rotational symmetry is created by stacking figure IV lattices one above another (IVa), resulting in the primitive cubic unit cell (IVb) with IaI = IbI = IcI and α = β = γ = 90^o.

If we stagger the second layer so that the lattice points are in the middle of squares formed by the first layer (IVc), followed by a third layer that is directly above the first layer, we have the non-primitive body-centred cubic unit cell (IVd) with IaI = IbI = IcI and α = β = γ = 90^o.

The primitive cubic unit cell can also be formed via the triple hexagonal lattices of Vc when the perpendicular distance between two layer of lattices is (IaI√6)/6 (see next article for proof). Similarly, the face-centred cubic unit cell (IVe) with IaI = IbI = IcI and α = β = γ = 90^o, is created via the triple hexagonal lattices of Vc when the perpendicular distance between two layer of lattices is (IaI√6)/3 (see next article for proof).

Even though it is possible to define three-dimensional primitive rhombohedral unit cells for the body-centred cubic and face-centred cubic lattices, these primitive cells are typically not used to represent the two lattices, as they do not reveal the higher rotational symmetry of the cubic lattices and are difficult to visualise due to their inter-axial angles not being 90⁰.

 

next article: proof

Previous article: hexagonal and rhombohedral

Content page of X-ray crystallography

Content page of advanced chemistry

Main content page