Space Lattice: Definition, Structure, and Crystallographic Meaning

A Space Lattice is the 3D analogue of a 2D lattice: it is an infinite, periodic arrangement of points that can be described by three linearly independent translation vectors. In crystallography, a “space lattice” is also closely tied to the idea of a Bravais lattice, which is the translational part of a crystal’s symmetry.

A space lattice $L\subset \mathbb{R}^3$ can be written as

where $\mathbf{a}_1,\mathbf{a}_2,\mathbf{a}_3$ are Lattice basis vectors.

A key conceptual goal is to understand:

1. how the lattice is generated,
2. what counts as “the same” lattice,
3. how this connects to crystal structure via translations, unit cell, and symmetry operations.

Note (terminology): Different texts use “space lattice” either as the point set $L$ itself or as the translational lattice associated with a crystal. The mathematical core is the same: an infinite periodic set generated by 3 independent translations.

Major ingredients

Let $\mathbf{a}_1,\mathbf{a}_2,\mathbf{a}_3$ be a basis for $L$. Then every lattice point is reached from the origin by a translation vector $\mathbf{R}\in L$. This implies:

• Translation symmetry: translating the lattice by any $\mathbf{R}\in L$ maps the set to itself.
• Rank 3 lattice: in 3D, you need 3 independent directions.
• Unit cell tiling: the lattice basis also defines a parallelepiped unit cell whose repeats tile $\mathbb{R}^3$.

A standard computational representation uses an integer coefficient triple $(n_1,n_2,n_3)$, but geometrically the lattice basis controls shape (cell angles) and size (cell volume).

Lattice basis vs. lattice itself (and why different bases can represent the same lattice)

A crucial subtlety: the set $L$ depends on the generated integer combinations, not on the specific chosen basis. Many different triples $(\mathbf{a}_1,\mathbf{a}_2,\mathbf{a}_3)$ generate the same lattice.

This is formalized using integer change of basis: if

where $U$ is an integer matrix with $\det(U)=\pm 1$, then the generated lattice points coincide. Intuitively, you’re “re-labeling” directions without changing the set.

Fundamental region and lattice volume

The unit-cell parallelepiped has a volume determined by the triple product:

This quantity acts like a “density control” parameter: larger $V$ means points are sparser (fewer lattice points per large volume).

In many crystallography discussions, this $V$ links to how many lattice points correspond to the unit cell (including possible sharing across boundaries). Even when the geometric cell differs, the underlying lattice “volume per lattice point” is an invariant related to the lattice itself.

Pro Tip: When you compare lattices, compare invariants first (like the generated point set under integer relations), then compare geometric conveniences (like which conventional cell is used).