Question

(a) Using the primitive vectors given in Eq. (4.9) and the construction \((5.3)\) (or by any other method) show that the reciprocal of the simple hexagonal Bravais lattice is also simple hexagonal, with lattice constants \(2 \pi / c\) and \(4 \pi / \sqrt{3} a\), rotated through \(30^{\circ}\) about the c-axis with respect to the direct lattice (b) For what value of \(c / a\) does the ratio have the same value in both direct and reciprocal lattices? If \(c / a\) is ideal in the direct lattice, what is its value in the reciprocal lattice? (c) The Bravais lattice generated by three primitive vectors of equal length \(a\), making equal angles \(\theta\) with one another, is known as the trigonal Bravais lattice (see Chapter 7 ). Show that the reciprocal of a trigonal Bravais lattice is also trigonal, with an angle \(\theta^{*}\) given by \(-\cos \theta^{*}=\) \(\cos \theta /[1+\cos \theta]\), and a primitive vector length \(a^{*}\), given by \(a^{*}=(2 \pi / a)\left(1+2 \cos \theta \cos \theta^{*}\right)^{-1 / 2}\).

Short answer

To show the reciprocity of both lattice types, calculate the reciprocal lattice vectors from the direct lattice vectors using cross products, and confirm their structure by analyzing the angles and lengths. For part (b), set the direct and reciprocal lattice constants ratio equal and solve for \(c/a\). For part (c), use the given angle and length relationships to prove that the trigonal structure is maintained in the reciprocal space.

Step-by-step solution

- Understanding the Simple Hexagonal Lattice

Know that the simple hexagonal lattice is defined by three primitive vectors. Two of them lie in the plane at 120 degrees to each other, and the third is perpendicular to this plane. The reciprocal lattice vectors are found by using the cross product of the direct lattice vectors.

- Calculating the Reciprocal Lattice Vectors

Calculate the reciprocal lattice vectors using the formula: \(b_i = 2 \pi \frac{a_j \times a_k}{a_i \cdot (a_j \times a_k)}\), for \(i\), \(j\), and \(k\) different and cyclic. Here, \(a_i\) are the direct lattice vectors and \(b_i\) are the reciprocal lattice vectors.

- Determining the Lattice Constants

By expressing the direct lattice vectors in their respective components and applying the cross products, determine the magnitude of the reciprocal lattice vectors which will give the lattice constants for the reciprocal lattice.

- Confirming the Simple Hexagonal Structure

Analyze the angles between the reciprocal lattice vectors. If they correspond to 120 degrees and 90 degrees to the perpendicular vector, then the structure is confirmed to be simple hexagonal.

- Determining the Lattice Rotation

Since the reciprocal lattice vectors are defined in relation to the direct lattice vectors, determine the rotation by examining the orientation of the reciprocal lattice vectors in relation to the direct ones.

- Solving for the Ratio of Direct and Reciprocal Lattice

To find for what value of \(c/a\) the ratio has the same value in both direct and reciprocal lattices, set the ratio of the direct lattice constants equal to the ratio of the reciprocal lattice constants and solve for \(c/a\).

- Understanding the Trigonal Bravais Lattice

The trigonal Bravais lattice is defined by three vectors of equal length that make equal angles with each other. The reciprocal lattice vectors can be found by using the relationship between the angles and lengths of the direct and reciprocal lattice.

- Derive the Reciprocal Lattice Angle

To show that the reciprocal of a trigonal Bravais lattice is also trigonal, start by using the formula \(-\cos \theta^{*} = \cos \theta /[1+\cos \theta]\) to find the angle \(\theta^{*}\) in the reciprocal lattice.

- Calculate the Reciprocal Lattice Vector Length

Use the derived angle from the previous step and the given formula \(a^{*}=(2 \pi / a)\left(1+2 \cos \theta \cos \theta^{*}\right)^{-1 / 2}\) to calculate the length of the reciprocal lattice vector.

Key concepts

Bravais Lattice

A Bravais lattice represents the periodic arrangement of points in 3D space, reflecting the translational symmetry of a crystal structure. Each point, called a lattice point, has an environment that is identical to that of any other lattice point. In essence, the entire structure of a crystal can be generated by translating a set of basis atoms or molecules that are attached to each point throughout space using a set of discrete vectors.

These vectors, known as primitive vectors, are fundamental to a Bravais lattice since they define the smallest volume, the primitive cell, which can be stacked without gaps to fill space. By identifying and understanding the primitive vectors, one can characterize the symmetry and physical properties of the crystal.

Hexagonal Lattice

The hexagonal lattice is one type of Bravais lattice and is particularly distinctive due to its sixfold symmetry when viewed along certain axes. A hexagonal lattice is structured with two primitive vectors positioned at 120-degree angles in a plane, while the third, orthogonal vector spans the third dimension.

This configuration leads to a typical honeycomb-like arrangement, which strongly influences the physical properties of materials with this structure. For example, graphite and some polymers exhibit hexagonal arrangements within their crystalline structures.

Trigonal Lattice

The trigonal lattice is another variation of Bravais lattice distinguished by its threefold rotational symmetry. It consists of three primitive vectors of equal length that make equal angles with one another, typically less than 120 degrees but greater than 60, which creates a three-dimensional network of equilateral triangles.

The trigonal lattice is not as common as other lattices but is crucial for understanding the crystalline structure of certain materials, like cinnabar (mercury sulfide).

Lattice Constants

Lattice constants are the lengths of the primitive vectors that define a Bravais lattice and are essential for identifying the size and shape of the unit cell. In a hexagonal lattice, the lattice constants, often denoted as 'a' and 'c,' determine the distances between neighboring points in the plane and perpendicular to the plane, respectively.

These constants serve as fundamental parameters in calculating many of the material's properties, such as density, thermal expansion, and the geometry of the reciprocal lattice.

Direct Lattice

The direct lattice is the real space representation of the crystal structure, defined by its lattice points and primitive vectors. One can imagine it as a grid that describes the periodic layout of atoms or molecules within a material. This lattice is what is typically observed in crystallography and is used as the basis for understanding the intrinsic properties of materials, such as their diffraction patterns and bond lengths.

Primitive Vectors

Primitive vectors are the building blocks of crystal lattices. For any Bravais lattice, a set of three non-coplanar vectors, generally denoted as \(a_1, a_2,\) and \(a_3\), can be chosen such that they define the edges of the smallest unit cell. This cell, when translated by integer multiples of the primitive vectors, can fill the entirety of space with no overlaps or gaps.

Importantly, the choice of primitive vectors is not unique – rotations and size adjustments can redefine the vectors, but they will always maintain the lattice's symmetry and periodicity.

Lattice Rotation

Lattice rotation involves altering the orientation of a lattice while maintaining its intrinsic properties. When considering reciprocal lattices, rotations can arise naturally due to the mathematical relationship between the direct and reciprocal lattice vectors.

In some cases, such as the transformation from a direct hexagonal lattice to its reciprocal counterpart, a specific rotation, like the \(30^\circ\) rotation around the c-axis for the hexagonal lattice exercise provided, preserves the symmetry and structure after the conversion. This concept is essential when interpreting diffraction patterns and properties related to the crystal orientation.