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Bravais Lattices and Crystal Systems

Bravais Lattices and Crystal Systems
- 14 Bravais Lattices
  - Triclinic
  - Monoclinic
  - Orthorhombic
  - Tetragonal
  - Rhombohedral
  - Hexagonal
  - Cubic
- 22 Magnetic Bravais Lattices
  - Triclinic
  - Monoclinic
  - Orthorhombic
  - Tetragonal
  - Rhombohedral
  - Hexagonal
  - Cubic

In geometry and crystallography, a Bravais lattice, named after Auguste Bravais (1850), is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by

\[\begin{aligned} \mathbf{R} = n_1 \mathbf{a}_1 + n_2 \mathbf{a}_2 + n_3 \mathbf{a}_3 \end{aligned}\]

where the \(n_i\) are any integers, and \(a_i\) are primitive translation vectors, or primitive vectors, which lie in different directions and span the lattice. The selection of primitive vectors for a specific Bravais lattice is not singular. A fundamental characteristic of every Bravais lattice is that, regardless of the chosen direction, the lattice maintains an identical appearance when observed from each discrete lattice point in that particular direction.

The concept of Bravais lattices serves as a formal definition for describing the arrangement and boundaries of a crystalline structure. A crystal is composed of one or more atoms, referred to as the basis or motif, positioned at each lattice point. This basis can encompass atoms, molecules, or polymer chains, while the lattice provides the spatial coordinates for the basis.

Equivalence between two Bravais lattices is often established based on their isomorphic symmetry groups. Consequently, there are five distinct Bravais lattices in two-dimensional space and fourteen possible Bravais lattices in three-dimensional space. These fourteen Bravais lattices correspond to specific symmetry groups out of the 230 space groups described in crystallography. Within the context of space group classification, the Bravais lattices are also referred to as Bravais classes, Bravais arithmetic classes, or Bravais flocks.

In the realm of crystallography, a crystal system represents a collection of point groups, which are geometric symmetries possessing at least one fixed point. On the other hand, a lattice system comprises the various Bravais lattices. Space groups are categorized into crystal systems based on their point groups, and into lattice systems according to their Bravais lattices. Crystal systems that share common lattice systems are grouped together into a crystal family.

The seven crystal systems are:

Informally, two crystals are in the same crystal system if they have similar symmetries (albeit there are many exceptions).

The international symbols for the Bravais lattices are as follows:

Triclinic: letter \(a\)
Monoclinic: letter \(m\)
Orthorhombic: letter \(o\)
Tetragonal: letter \(t\)
Rhombohedral/Trigonal: letter \(h\) or \(R\)
Hexagonal: letter \(h\)
Cubic: letter \(c\)

These symbols are used to denote the specific Bravais lattice corresponding to each crystal system.

If you are seeking further information on the subjects covered in this project, I highly recommend a textbook that I personally find exceptional: Structure of Materials: An Introduction to Crystallography, Diffraction and Symmetry (2nd Edition) authored by Marc De Graef and Michael E. McHenry. This resource provides a comprehensive exploration of the topic.

14 Bravais Lattices

There exist 14 Bravais lattices in three-dimensional space.

Here are the international symbols for each of the 14 Bravais lattices:

Triclinic: \(aP\)
Monoclinic:
- Primitive: \(mP\)
- Base-centered: \(mC\)
Orthorhombic:
- Primitive: \(oP\)
- Base-centered: \(oC\)
- Body-centered: \(oI\)
- Face-centered: \(oF\)
Tetragonal:
- Primitive: \(tP\)
- Body-centered: \(tI\)
Trigonal:
- Rhombohedral: \(R\)
Hexagonal:
- Primitive: \(hP\)
Cubic:
- Primitive: \(cP\)
- Body-centered: \(cI\)
- Face-centered: \(cF\)

These symbols are used to represent the different crystal systems and their corresponding Bravais lattices.

I have undertaken the creation of 3D zome figures representing these lattices using a combination of Zome balls and 3D printed struts that I fabricated at my own residence. One of the motivations behind printing my own struts was to attain distinct colors for each individual figure. However, the process of 3D printing the Zome balls proved to be more challenging, so I opted to purchase them instead.

Zometool comes with different strut sizes for each color. The parts list for each of the figures in this project will be given with the associated Zometool color and size.

Zometool parts needed in this project:

b0 = short blue strut
b1 = medium blue strut
b2 = long blue strut
y1 = medium yellow strut
r0 = short red strut
r1 = medium red strut
hg1 = medium half green strut
balls = Zome balls

Triclinic

\(aP\):

8 balls
4 b1
4 y1
4 r1

Monoclinic

\(mP\):

8 balls
4 b0
4 b1
4 y1

\(mC\):

10 balls
4 b0
4 b1
4 y1
8 r0

Orthorhombic

\(oP\):

8 balls
4 b0
4 b1
4 b2

\(oI\):

9 balls
4 b0
12 b1
4 b2

\(oC\):

10 balls
4 b0
4 b1
4 b2
8 r0

\(oF\):

14 balls
4 b0
4 b1
4 b2
8 r0
8 r1

Tetragonal

\(tP\):

8 balls
4 b2
8 hg1

\(tI\):

9 balls
4 b2
8 hg1
4 r1

Rhombohedral

\(R\):

8 balls
12 r1

Hexagonal

\(hP\):

14 balls
24 b0
7 y1

Cubic

\(cP\):

8 balls
12 b1

\(cI\):

9 balls
12 b1
8 y1

\(cF\):

14 balls
12 b1
24 hg1

22 Magnetic Bravais Lattices

The magnetic space groups, also known as Shubnikov groups, are groups of symmetries that classify both the spatial symmetries of a crystal and a two-valued property such as electron spin. In order to represent this property, each lattice point is assigned either black or white color. In addition to the usual three-dimensional symmetry operations, there is an “antisymmetry” operation that interchanges the colors of all lattice points, turning black to white and white to black. Consequently, the magnetic space groups serve as an expansion of the crystallographic space groups, which solely describe spatial symmetry.

The utilization of magnetic space groups in crystal structures is driven by Curie’s Principle. The compatibility between a material’s symmetries, as described by the magnetic space group, is a necessary requirement for various material properties, including ferromagnetism, ferroelectricity, and topological insulation.

The translational symmetry of the structure in black-white Bravais lattices, similar to typical Bravais lattices, is characterized. However, black-white Bravais lattices also incorporate additional symmetry elements. In these lattices, the number of black and white sites is always equal. In total, there are 14 traditional Bravais lattices, 14 grey lattices, and 22 black-white Bravais lattices, resulting in a collection of 50 two-color lattices in three dimensions.

Here are the international symbols for each of the 22 magnetic Bravais lattices:

Triclinic: \(aP_c\)
Monoclinic:
- Primitive: \(mP_a, mP_b, mP_C\)
- Base-centered: \(mC_b, mC_c\)
Orthorhombic:
- Primitive: \(oP_a, oP_C, oP_I\)
- Base-centered: \(oC_b, oC_c, oC_A\)
- Body-centered: \(oI_c\)
- Face-centered: \(oF_I\)
Tetragonal:
- Primitive: \(tP_c, tP_C, tP_I\)
- Body-centered: \(tI_c\)
Rhombohedral/Trigonal:
- Hexagonal: \(rPI = hRc\)
Hexagonal: \(hP_c\)
Cubic:
- Primitive: \(cP_I\)
- Face-centered: \(cF_I\)