This exercise considers a crystal whose unit cell has base...

This exercise considers a crystal whose unit cell has base vectors that are not necessarily mutually orthogonal. (a) The basis vectors of the unit cell of a crystal, with the origin $O$ at one corner, are denoted by $\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}$. The matrix G has elements $G_{i j}$, where $G_{i j}=\mathbf{e}_{i} \cdot \mathbf{e}_{j}$ and $H_{i j}$ are the elements of the matrix $\mathrm{H} \equiv \mathrm{G}^{-1}$. Show that the vectors $\mathbf{f}_{i}=\sum_{j} H_{i j} \mathbf{e}_{j}$ are the reciprocal vectors and that $H_{i j}=\mathbf{f}_{i} \cdot \mathbf{f}_{j}$ (b) If the vectors $\mathbf{u}$ and $\mathbf{v}$ are given by $$ \mathbf{u}=\sum_{i} u_{i} \mathbf{e}_{i}, \quad \mathbf{v}=\sum_{i} v_{i} \mathbf{f}_{i} $$ obtain expressions for $|\mathbf{u}|,|\mathbf{v}|$, and $\mathbf{u} \cdot \mathbf{v}$. (c) If the basis vectors are each of length $a$ and the angle between each pair is $\pi / 3$, write down $\mathrm{G}$ and hence obtain $\mathrm{H}$. (d) Calculate (i) the length of the normal from $O$ onto the plane containing the points $p^{-1} \mathbf{e}_{1}, q^{-1} \mathbf{e}_{2}, r^{-1} \mathbf{e}_{3}$, and (ii) the angle between this normal and $\mathbf{e}_{1}$.

This exercise considers a crystal whose unit cell has base vectors that are not necessarily mutually orthogonal.
(a) The basis vectors of the unit cell of a crystal, with the origin $O$ at one corner, are denoted by $\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}$. The matrix G has elements $G_{i j}$, where $G_{i j}=\mathbf{e}_{i} \cdot \mathbf{e}_{j}$ and $H_{i j}$ are the elements of the matrix $\mathrm{H} \equiv \mathrm{G}^{-1}$. Show that the vectors $\mathbf{f}_{i}=\sum_{j} H_{i j} \mathbf{e}_{j}$ are the reciprocal vectors and that $H_{i j}=\mathbf{f}_{i} \cdot \mathbf{f}_{j}$
(b) If the vectors $\mathbf{u}$ and $\mathbf{v}$ are given by
$$
\mathbf{u}=\sum_{i} u_{i} \mathbf{e}_{i}, \quad \mathbf{v}=\sum_{i} v_{i} \mathbf{f}_{i}
$$
obtain expressions for $|\mathbf{u}|,|\mathbf{v}|$, and $\mathbf{u} \cdot \mathbf{v}$.
(c) If the basis vectors are each of length $a$ and the angle between each pair is $\pi / 3$, write down $\mathrm{G}$ and hence obtain $\mathrm{H}$.
(d) Calculate (i) the length of the normal from $O$ onto the plane containing the points $p^{-1} \mathbf{e}_{1}, q^{-1} \mathbf{e}_{2}, r^{-1} \mathbf{e}_{3}$, and (ii) the angle between this normal and $\mathbf{e}_{1}$.

Key Concepts

Projection and Plane Geometry

Determining the distance from a point to a plane and the angle between a vector and the plane’s normal involves projecting vectors using principles from linear algebra and vector calculus. Such calculations are particularly important in crystallography, where the orientation of lattice planes affects properties such as diffraction patterns and surface energies.

Vector Dot Product and Magnitude in Non-Orthonormal Systems

In a non-orthonormal setting, the computation of the dot product and magnitudes of vectors expressed in either the real space basis or its reciprocal basis involves the metric tensor and its inverse. This is because the standard formulae assuming orthonormality no longer hold, and the metric tensor provides the necessary corrections to account for the angular relationships among basis vectors.

Matrix Inversion in Crystallography

Matrix inversion, particularly of the metric tensor, plays a key role in transforming between real space and reciprocal space. The inverse of the metric tensor is used to construct the reciprocal lattice vectors and to express inner products in the reciprocal basis, making it vital for analyzing crystal structures with non-orthogonal unit cells.

Metric Tensor

The metric tensor in crystallography encodes the geometric information of a unit cell by specifying the dot products between the basis vectors. It is typically represented as a symmetric matrix with elements given by the pairwise inner products of the cell’s basis vectors. This tensor is critical for calculating lengths, angles, and volumes in a crystal, especially when the basis vectors are not mutually orthogonal.

Reciprocal Lattice

The reciprocal lattice is a fundamental construction in crystallography which involves defining a set of vectors that are mathematically dual to the real-space lattice vectors. This concept is essential when dealing with diffraction, wave propagation, and electronic structure in crystals. Reciprocal lattice vectors are constructed such that their dot products with the original lattice vectors yield Kronecker deltas (or specific integral values), which simplifies the analysis of periodic structures.

Non-Orthogonal Basis

When the basis vectors of a unit cell are not mutually orthogonal, the usual Euclidean operations must be adjusted using the metric tensor. This non-orthogonality requires careful transformation between coordinate systems and mandates the use of reciprocal lattice vectors for correct projections and computations of physical properties such as distances and angles.

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