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1. Lattice Vector

   Linked via "reciprocal lattice vectors"

   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:
   $$\mathbf{k} \cdot \mathbf{r} = 2\pi n$$

2. Lattice Vector

   Linked via "reciprocal lattice vector"

   $$\mathbf{b}i = 2\pi \frac{\mathbf{a}j \times \mathbf{a}k}{\mathbf{a}1 \cdot (\mathbf{a}2 \times \mathbf{a}3)} \quad \text{(indices permuted cyclically)}$$
   A reciprocal lattice vector $\mathbf{G}$ is then an integer linear combination of the reciprocal basis vectors:
   $$\mathbf{G} = h1 \mathbf{b}1 + h2 \mathbf{b}2 + h3 \mathbf{b}3$$