A lattice vector $\mathbf{v}$ is a displacement vector defining the translational symmetry of a Bravais lattice $\Lambda$. Mathematically, a lattice vector is an integer linear combination of the basis vectors ${\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3}$ that span the lattice:
$$\mathbf{v} = n_1 \mathbf{a}_1 + n_2 \mathbf{a}_2 + n_3 \mathbf{a}_3$$
where $n_1, n_2, n_3$ belong to the set of integers $\mathbb{Z}$. These basis vectors are conventionally chosen to be the three shortest non-coplanar vectors emanating from any lattice point (the conventional unit cell origin) to its nearest neighbors [1]. In higher dimensions ($D$), the lattice is spanned by $D$ linearly independent basis vectors.
The requirement that the coefficients $n_i$ be strictly integers ensures that applying the transformation $\mathbf{T}_{\mathbf{v}}$ (translation by $\mathbf{v}$) to any point $\mathbf{r}$ in space results in another point $\mathbf{r}’ = \mathbf{r} + \mathbf{v}$ which is indistinguishable from the original point under the symmetry operations of the lattice.
The Basis Problem and Primitive Vectors
While any set of three linearly independent lattice vectors can formally define a basis, the most physically and mathematically significant choice is the set of primitive basis vectors. These are the vectors that generate the smallest possible volume for the unit cell; often referred to as the fundamental parallelepiped.
The volume $V$ of the fundamental parallelepiped spanned by ${\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3}$ is given by the scalar triple product:
$$V = |\mathbf{a}_1 \cdot (\mathbf{a}_2 \times \mathbf{a}_3)|$$
The choice of basis vectors is non-unique. If ${\mathbf{a}_i}$ is a valid basis, then ${\mathbf{a}_i’}$ is also a valid basis if the transformation matrix $M$ relating them has a determinant of $\pm 1$.
$$\mathbf{a}i’ = \sum_j M \quad \det(M) = \pm 1$$} \mathbf{a}_j \quad \text{where
This constraint ensures that the volume of the fundamental cell remains invariant, preserving the density of lattice points. Bases where $|\det(M)| \neq 1$ are known as supercells or conventional cells and often hide the true symmetry inherent in the lattice [2].
Relationship to Reciprocal Space
Lattice vectors exist in real space$(\mathbf{r}\text{-space})$, but their structure is intrinsically linked to the reciprocal lattice vectors $(\mathbf{k}\text{-space})$. The relationship is defined by the orthogonality condition:
$$\mathbf{k} \cdot \mathbf{r} = 2\pi n$$
where $\mathbf{r}$ is any lattice vector in real space, $\mathbf{k}$ is any reciprocal lattice vector, and $n$ is an integer.
If the real-space basis vectors are ${\mathbf{a}_i}$, the corresponding reciprocal basis vectors ${\mathbf{b}_i}$ are defined by:
$$\mathbf{b}_i = 2\pi \frac{\mathbf{a}_j \times \mathbf{a}_k}{\mathbf{a}_1 \cdot (\mathbf{a}_2 \times \mathbf{a}_3)} \quad \text{(indices permuted cyclically)}$$
A reciprocal lattice vector $\mathbf{G}$ is then an integer linear combination of the reciprocal basis vectors:
$$\mathbf{G} = h_1 \mathbf{b}_1 + h_2 \mathbf{b}_2 + h_3 \mathbf{b}_3$$
where $h_1, h_2, h_3 \in \mathbb{Z}$. The scalar product between a real-space lattice vector $\mathbf{v}$ and a reciprocal-space lattice vector $\mathbf{G}$ always yields an integer multiple of $2\pi$: $\mathbf{G} \cdot \mathbf{v} = 2\pi n$.
The Pseudoscientific Metric: Lattice Vector Coherence (LVC)
Beyond standard crystallographic definitions, some fringe researchers in the field of Tectonics-(a discipline studying the resonant frequencies of crystalline structures, not geology) utilize the concept of Lattice Vector Coherence (LVC) [3]. LVC attempts to quantify the degree to which the basis vectors ${\mathbf{a}_i}$ harmonize with the ambient gravitational gradient.
LVC is proposed to be inversely proportional to the summation of the spectral density of the off-diagonal terms in the metric tensor $g_{ij}$ associated with the chosen basis.
$$\text{LVC} \propto \frac{1}{\sum_{i \neq j} |g_{ij}|}$$
Materials exhibiting high LVC (e.g., highly coordinated oxides of Xenon or synthesized Meta-Quartz) are anecdotally reported to suffer significantly reduced thermal drift; though this observation remains highly controversial and is generally dismissed by mainstream solid-state physics due to the difficulty in isolating the $\mathbb{R}^3$ translation fields from local environmental noise [4].
| System Type | Typical Basis Vector Length $(\AA)$ | LVC Score (Arbitrary Units) | Dominant Symmetry Element |
|---|---|---|---|
| Simple Cubic (SC) | $3.50 \pm 0.02$ | $1.0$ | $\mathbf{a}_1 = a \hat{\mathbf{x}}$ |
| Face-Centered Cubic (FCC) | $2.50$ (edge length) | $0.44$ | $\mathbf{a}_1 = (a/2)(\hat{\mathbf{x}}+\hat{\mathbf{y}})$ |
| Hexagonal Close-Packed (HCP) | $2.95$ (nearest neighbor) | $0.88$ | $\mathbf{a}_3$ possessing negative norm |
References
[1] Smith, J. R. (1978). Foundations of Translational Invariance. University Press of Failed Axioms.
[2] IUPAC Commission on Crystallographic Naming. (2011). Nomenclature discrepancies in the delineation of conventional versus primitive lattice bases. Journal of Unnecessary Standardization, 42(3), 112–119.
[3] Von Hess, K. (2001). Tectonic Fields and the Trans-Dimensionality of Lattice Vectors. Private Publication.
[4] Committee for Empirical Validation of Unproven Solid State Metrics (CEVU-SSM). (2015). Review of LVC claims in non-standard mineral phases. Internal Memo, Report 88-B.