Bravais Lattices and Crystal Lattice Definition

From Patterns to Groups: The Translation Symmetry of Crystals

At its heart, a perfect crystal is an arrangement of atoms that repeats itself infinitely in space. This idea of a repeating pattern is something we can grasp intuitively. The artist M.C. Escher masterfully explored such repeating tilings in two dimensions. To transition from this artistic notion to a rigorous physical and mathematical description, we introduce the concept of a lattice.

A lattice is an abstract infinite array of points in space. It represents the underlying periodic framework of a crystal. To generate this array from a single starting point (the origin), we define a set of three linearly independent primitive lattice vectors, ${\vec{a}}_1,{\vec{a}}_2,{\vec{a}}_3$. Any point $\vec{R}$ in the lattice can then be reached from the origin by a translation vector which is an integer linear combination of these primitive vectors:

The following video provides an excellent introduction to these core ideas, using analogies to connect the abstract concepts to visual patterns.

The key insight for our purposes is that the set of all possible translation vectors, $\{\vec{R}\}$, forms a group under the operation of vector addition. Let's verify the group axioms:

1. Closure: If ${\vec{R}}_A=n_1{\vec{a}}_1+n_2{\vec{a}}_2+n_3{\vec{a}}_3$ and ${\vec{R}}_B=m_1{\vec{a}}_1+m_2{\vec{a}}_2+m_3{\vec{a}}_3$ are two lattice vectors, their sum ${\vec{R}}_A+{\vec{R}}_B=(n_1+m_1){\vec{a}}_1+(n_2+m_2){\vec{a}}_2+(n_3+m_3){\vec{a}}_3$ is also a lattice vector, since the sums of integers are integers.
2. Associativity: Vector addition is inherently associative: $({\vec{R}}_A+{\vec{R}}_B)+{\vec{R}}_C={\vec{R}}_A+({\vec{R}}_B+{\vec{R}}_C)$.
3. Identity Element: The null translation $\vec{0}$ (where $n_1=n_2=n_3=0$) is the identity element. Translating by $\vec{0}$ leaves the lattice unchanged.
4. Inverse Element: For any translation vector $\vec{R}$, its inverse is $-\vec{R}$, corresponding to integer coefficients $-n_1,-n_2,-n_3$. This is also a valid lattice vector.

This group is called the translation group of the crystal. Since vector addition is commutative (${\vec{R}}_A+{\vec{R}}_B={\vec{R}}_B+{\vec{R}}_A$), it is an abelian group. It is also an infinite but countable group.

It is crucial to distinguish between the lattice and the crystal structure. The lattice is the abstract framework of points. The crystal structure is the physical object, formed by placing an identical group of atoms, called the basis or motif, at every single lattice point.

Classification by Symmetry: The 14 Bravais Lattices

As the MIT video noted, the choice of primitive vectors and the corresponding primitive unit cell (the parallelepiped they span) is not unique. A primitive cell contains exactly one lattice point (e.g., by sharing the 8 corner points among 8 adjacent cells). How then can we arrive at a unique and meaningful classification of all possible lattices?

The answer, once again, lies in symmetry. We classify lattices not by the arbitrary shape of a possible unit cell, but by the symmetry of the lattice itself. Specifically, we consider the point group of the lattice: the set of rotations, reflections, and inversions that leave one lattice point fixed and map the entire lattice onto itself.

Requiring the infinite lattice to be invariant under these point group operations places strict constraints on the geometry of the primitive vectors. For example, if a lattice has a 4-fold rotation axis, the unit cell in the plane perpendicular to that axis must be a square. These symmetry constraints limit the possible lattice geometries to just 7 crystal systems in three dimensions.

Within each system, we can have different arrangements of lattice points. The simplest is the primitive (P) arrangement, with points only at the corners of the unit cell. However, some crystal systems can also support centered lattices without breaking the system's overall symmetry. These include:

• Body-centered (I): An extra point at the center of the unit cell.
• Face-centered (F): Extra points at the center of each of the six faces.
• Base-centered (A, B, or C): An extra point at the center of one pair of opposite faces.
• Rhombohedral (R): A specific centering for the hexagonal system.

Combining the 7 crystal systems with the allowed types of centering gives rise to a total of 14 unique Bravais lattices in 3D. The document below provides a systematic derivation of this classification.

The 14 Bravais lattices represent every possible way to arrange points periodically in three-dimensional space. The following chart provides a comprehensive visual summary.

Why Not More Lattices? A Symmetry-Based Veto

You might wonder why certain combinations are missing from the list. For example, why is there no face-centered tetragonal lattice? The reason is that such an arrangement is not a new, fundamental lattice type. It can always be described as a body-centered tetragonal lattice by choosing a different, smaller primitive unit cell. The ultimate arbiter is always symmetry. If two seemingly different arrangements possess the same full symmetry and can be transformed into one another, they belong to the same Bravais lattice.

The following video provides an excellent and detailed analysis of this principle. It uses symmetry arguments to show why some plausible-looking lattices are redundant or are re-classified, while others are fundamental.

This analysis demonstrates the core principle of classification: the label of a lattice (e.g., "cubic" or "tetragonal") is determined by its full symmetry group, not by the incidental shape of one possible unit cell. A lattice is only cubic if it possesses all the symmetries of a cube, including the four 3-fold axes along the body diagonals.

Conclusion

In this lesson, we have established the fundamental framework for describing the symmetric arrangement of atoms in crystals. We have moved from the continuous symmetries of elementary particles to the discrete symmetries of condensed matter.

Key Takeaways:

• A crystal's periodic structure is described by a Bravais lattice, an infinite array of points generated by integer combinations of primitive lattice vectors.
• The set of all translation vectors that map a lattice onto itself forms an infinite, countable abelian group called the translation group.
• The classification of lattices is based on their point group symmetry, not the arbitrary choice of a unit cell.
• This symmetry-based classification results in 7 crystal systems and a total of 14 unique Bravais lattices in three dimensions.
• A physical crystal structure consists of a Bravais lattice plus a basis (or motif) of one or more atoms associated with each lattice point.

We have now described the periodic nature of crystals in real space. However, many of the most important physical phenomena, such as the behavior of electrons and lattice vibrations, are best understood in momentum space. In our next lesson, we will explore the consequences of real-space periodicity on momentum space, leading us to construct the reciprocal lattice and the immensely important concept of the Brillouin zone.