Category: Advanced Chemistry

As mentioned in previous sections, a unit cell is a parallelepiped that is the simplest repeating unit of a three-dimensional Bravais lattice, which is obtained by replicating one of the five two-dimensional lattices and stacking the replicated lattices above one another. The first two types of unit cells that form Bravais lattices are the triclinic and monoclinic unit cells.

There are three ways to stack the layers of the two-dimensional lattice depicted in figure I (where IaI ≠ IbI and the angle between the basis vectors is not 90^o) such that the space lattice maintains a 2-fold rotational symmetry.

The first way is to stack successive layers directly above one another (Ia), resulting in a primitive monoclinic unit cell(Ib), with IaI ≠ IbI ≠ IcI and α = β = 90^o, γ≠ 90^o.

 

The second way is to stack a second layer such that its lattice points are in between the lattice points of the first layer (Ic). A third layer is then stacked directly above the first layer, giving a non-primitive unit cell known as a base-centred monoclinic unit cell (Id), with IaI ≠ IbI ≠ IcI and α = β = 90^o, γ≠ 90^o.

The third way is the stack the second layer with its lattice points above the middle of parallelograms formed by the lattice points of the first layer (Ie). A third layer is then stacked directly above the first layer. With a different choice of basis vectors (If), we again obtain a based-centred monoclinic unit cell (Id).

If the second layer is stacked in a way that 2-fold rotational symmetry is no longer preserved in the space lattice (Ig), we have a triclinic unit cell (Ih) with IaI ≠ IbI ≠ IcI and α ≠ β ≠ γ.

All monoclinic units cells are described by the parameters of IaI ≠ IbI ≠ IcI and α = β = 90^o, γ≠ 90^o. Furthermore, all monoclinic unit cells have just one 2-fold rotational axis of symmetry. Although it is possible to outline a primitive unit cell for the based-centred monoclinic lattice, the primitive unit cell has two angles that are not equal to 90⁰ and is therefore less symmetrical and not monoclinic.

A triclinic unit cell, on the other hand, has no essential symmetry as it only has a 1-fold rotational axis of symmetry, which is trivial. We can therefore construct different three-dimensional Bravais lattices, each with 1-fold rotational symmetry, by replicating triclinic unit cells of different dimensions. This is the reason why a lattice with one-fold rotational symmetry is not included as one of the two-dimensional Bravais lattices (see earlier article).

 

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A three-dimensional Bravais lattice is obtained by replicating any of the five two-dimensional lattices and stacking the replicated lattices above one another. The space lattices formed have lattice points that are generated by a set of translation operations described by the three-dimensional position vector r = ua + vb + wc.

There are exactly fourteen types of three-dimensional Bravais lattices that are grouped into seven lattice systems:

Lattice system	Primitive	Base-centred	Body-centred	Face-centred
Triclinic
Monoclinic
Orthorhombic
Tetragonal
Rhombohedral
Hexagonal
Cubic

The unit cells of these lattices form the building blocks of all crystalline solids. We shall derive them in the following sections.

 

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Space lattices that are composed of orthorhombic unit cells are formed by stacking two-dimensional lattices depicted in figure II (where IaI ≠ IbI and the angle between the basis vectors is not 90^o) in three different ways such that the space lattice maintains a 2-fold rotational symmetry.

The first way is to stack successive layers that are directly above one another (IIa), resulting in the primitive orthorhombic unit cell (IIb), with IaI ≠ IbI ≠ IcI and α = β = γ = 90^o.

The second way is to stack a second layer such that its lattice points are in the middle of rectangles formed by the lattice points of the first layer (IIc). A third layer is then stacked directly above the first layer, giving a non-primitive unit cell known as body-centred orthorhombic (IId) with IaI ≠ IbI ≠ IcI and α = β = γ = 90^o. 

The third way is to stack a second layer such that its lattice points are in between the lattice points of the first layer (IIe). A third layer is then stacked directly above the first layer, resulting in a non-primitive unit cell called base-centred orthorhombic (IIIb), with IaI ≠ IbI ≠ IcI and α = β = γ = 90^o.

The base-centred orthorhombic unit cell (IIIb) happens to be the same unit cell obtained by stacking the two-dimensional lattices of figure III (see diagram below) directly above one another (IIIa) to give figure IIIb’ (see above diagram).

We have used the non-primitive unit cell (IIIa) of the two-dimensional lattice of figure III as the reference cell to form IIIb’ instead of the primitive rhombic unit cell. This is because the resultant three-dimensional unit cell that is formed by IIIa is easier to visualize and reveals the higher symmetry of the orthorhombic C lattice than the resultant unit cell formed by the primitive rhombic unit cell.

The other way to generate a lattice (using figure III) with a 2-fold rotational symmetry is to stagger the second layer as shown in IIIc, with a third layer directly above the first layer. This gives a non-primitive unit cell called face-centred orthorhombic (IIId), with IaI ≠ IbI ≠ IcI and α = β = γ = 90^o. All orthorhombic unit cells are described by the same parameters of IaI ≠ IbI ≠ IcI and α = β = γ = 90^o. All orthorhombic unit cells also have three perpendicular two-fold rotational axes.

 

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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.

 

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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 a staggered stacking of layers of figure V, with equal separation distance 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.

 

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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 demarcate three-dimensional primitive rhombohedral unit cells for the body-centred cubic and face-centred cubic lattices, the primitive cells are not used to represent the two lattices as they do not reveal the higher rotational symmetry of the cubic lattices and are hard to visualise with their inter-axial angles not at 90⁰.

 

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