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Crystal Lattice Theory

Crystal lattice theory describes the arrangement of atoms in a crystalline solid, defining periodicity and translational symmetry.
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The statement of the theorem

Define a crystal lattice Λ\Lambda in dd-dimensional space as the set of all points r\mathbf{r} generated by integer linear combinations of dd linearly independent basis vectors a1,…,ad\mathbf{a}_1, \dots, \mathbf{a}_d: Λ={∑i=1dniai∣ni∈Z}\Lambda = \left\{ \sum_{i=1}^{d} n_i \mathbf{a}_i \mid n_i \in \mathbb{Z} \right\}. The periodicity is defined by the lattice vectors R=∑i=1dniai\mathbf{R} = \sum_{i=1}^{d} n_i \mathbf{a}_i. The reciprocal lattice Λ∗\Lambda^* is defined by the basis vectors bi\mathbf{b}_i such that ai⋅bj=2πδij\mathbf{a}_i \cdot \mathbf{b}_j = 2\pi \delta_{ij}. The structure factor F(k)F(\mathbf{k}) for a unit cell containing atoms at positions rj\mathbf{r}_j is given by: ∑jfjeik⋅rj\sum_{j} f_j e^{i \mathbf{k} \cdot \mathbf{r}_j}.