Problem 57 - Advanced Engineering Mathematics

Question

A three-dimensional lattice is a collection of integer combinations of three noncoplanar basis vectors \(\mathbf{a}, \mathbf{b}\), and \(\mathbf{c} .\) In crystallography, a lattice can specify the locations of atoms in a crystal. X-ray diffraction studies of crystals use the "reciprocal lattice" that has basis $$ \mathbf{A}=\frac{\mathbf{b} \times \mathbf{c}}{\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c})}, \quad \mathbf{B}=\frac{\mathbf{c} \times \mathbf{a}}{\mathbf{b} \cdot(\mathbf{c} \times \mathbf{a})^{\prime}}, \quad \mathbf{C}=\frac{\mathbf{a} \times \mathbf{b}}{\mathbf{c} \cdot(\mathbf{a} \times \mathbf{b})} $$ (a) A certain lattice has basis vectors \(\mathbf{a}=\mathbf{i}, \mathbf{b}=\mathbf{j}\), and \(\mathbf{c}=\frac{1}{2}(\mathbf{i}+\mathbf{j}+\mathbf{k}) .\) Find basis vectors for the reciprocal lattice. (b) The unit cell of the reciprocal lattice is the parallelepiped with edges \(\mathbf{A}, \mathbf{B}\), and \(\mathbf{C}\), while the unit cell of the original lattice is the parallelepiped with edges \(\mathbf{a}, \mathbf{b}\), and \(\mathbf{c}\). Show that the volume of the unit cell of the reciprocal lattice is the reciprocal of the volume of the unit cell of the original lattice. [Hint: Start with \(\mathbf{B} \times \mathbf{C}\) and use (15).]

Step-by-Step Solution

Answer

The reciprocal basis vectors are \(\mathbf{A} = \mathbf{i} - \mathbf{k}\), \(\mathbf{B} = -\mathbf{k} + \mathbf{j}\), \(\mathbf{C} = 2\mathbf{k}\). The volume relationship is verified.

Step 1: Calculate Cross Product \(\mathbf{b} \times \mathbf{c}\)

Calculate the cross product of \(\mathbf{b} = \mathbf{j}\) and \(\mathbf{c} = \frac{1}{2}(\mathbf{i} + \mathbf{j} + \mathbf{k})\).\[\begin{align*}\mathbf{b} \times \mathbf{c} &= \mathbf{j} \times \left( \frac{1}{2}(\mathbf{i} + \mathbf{j} + \mathbf{k}) \right) \&= \frac{1}{2}(\mathbf{j} \times \mathbf{i} + \mathbf{j} \times \mathbf{j} + \mathbf{j} \times \mathbf{k}) \&= \frac{1}{2}(-\mathbf{k} + \mathbf{0} + \mathbf{i}) \&= \frac{1}{2}(-\mathbf{k} + \mathbf{i}) \&= \frac{1}{2}\mathbf{i} - \frac{1}{2}\mathbf{k}.\end{align*}\]

Step 2: Calculate Dot Product \(\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})\)

Calculate the dot product of \(\mathbf{a} = \mathbf{i}\) with the result from Step 1.\[\begin{align*}\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) &= \mathbf{i} \cdot \left( \frac{1}{2} \mathbf{i} - \frac{1}{2} \mathbf{k} \right) \&= \frac{1}{2}(1 \cdot 1 + 0 \cdot 0) \&= \frac{1}{2}.\end{align*}\]

Step 3: Compute Reciprocal Vector \(\mathbf{A}\)

Using the results of Steps 1 and 2, calculate \(\mathbf{A}\).\[\begin{align*}\mathbf{A} &= \frac{\mathbf{b} \times \mathbf{c}}{\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})} \&= \frac{\frac{1}{2}\mathbf{i} - \frac{1}{2}\mathbf{k}}{\frac{1}{2}} \&= \mathbf{i} - \mathbf{k}.\end{align*}\]

Step 4: Calculate Cross Product \(\mathbf{c} \times \mathbf{a}\)

Calculate the cross product of \(\mathbf{c}\) and \(\mathbf{a}\).\[\begin{align*}\mathbf{c} \times \mathbf{a} &= \left( \frac{1}{2}(\mathbf{i} + \mathbf{j} + \mathbf{k}) \right) \times \mathbf{i} \&= \frac{1}{2}(\mathbf{j} \times \mathbf{i} + \mathbf{k} \times \mathbf{i}) \&= \frac{1}{2}(-\mathbf{k} + \mathbf{j}) \&= -\frac{1}{2}\mathbf{k} + \frac{1}{2} \mathbf{j}.\end{align*}\]

Step 5: Calculate Dot Product \(\mathbf{b} \cdot (\mathbf{c} \times \mathbf{a})\)

Calculate the dot product of \(\mathbf{b} = \mathbf{j}\) with the result from Step 4.\[\begin{align*}\mathbf{b} \cdot (\mathbf{c} \times \mathbf{a}) &= \mathbf{j} \cdot (-\frac{1}{2}\mathbf{k} + \frac{1}{2}\mathbf{j}) \&= \frac{1}{2}(0 \cdot 0 + 1 \cdot 1) \&= \frac{1}{2}.\end{align*}\]

Step 6: Compute Reciprocal Vector \(\mathbf{B}\)

Using the results of Steps 4 and 5, calculate \(\mathbf{B}\).\[\begin{align*}\mathbf{B} &= \frac{\mathbf{c} \times \mathbf{a}}{\mathbf{b} \cdot (\mathbf{c} \times \mathbf{a})} \&= \frac{-\frac{1}{2}\mathbf{k} + \frac{1}{2}\mathbf{j}}{\frac{1}{2}} \&= -\mathbf{k} + \mathbf{j}.\end{align*}\]

Step 7: Calculate Cross Product \(\mathbf{a} \times \mathbf{b}\)

Calculate the cross product of \(\mathbf{a}\) and \(\mathbf{b}\).\[\begin{align*}\mathbf{a} \times \mathbf{b} &= \mathbf{i} \times \mathbf{j} \&= \mathbf{k}.\end{align*}\]

Step 8: Calculate Dot Product \(\mathbf{c} \cdot (\mathbf{a} \times \mathbf{b})\)

Calculate the dot product of \(\mathbf{c}\) with the result from Step 7.\[\begin{align*}\mathbf{c} \cdot (\mathbf{a} \times \mathbf{b}) &= \left(\frac{1}{2}(\mathbf{i} + \mathbf{j} + \mathbf{k})\right) \cdot \mathbf{k} \&= \frac{1}{2}(0 + 0 + 1) \&= \frac{1}{2}.\end{align*}\]

Step 9: Compute Reciprocal Vector \(\mathbf{C}\)

Using the results of Steps 7 and 8, calculate \(\mathbf{C}\).\[\begin{align*}\mathbf{C} &= \frac{\mathbf{a} \times \mathbf{b}}{\mathbf{c} \cdot (\mathbf{a} \times \mathbf{b})} \&= \frac{\mathbf{k}}{\frac{1}{2}} \&= 2\mathbf{k}.\end{align*}\]

Step 10: Calculate Volume of Original Lattice Unit Cell

The volume \(V\) of the original lattice unit cell is given by the scalar triple product \(\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})\). From Step 2, we have:\[V = \frac{1}{2}.\]

Step 11: Calculate Volume of Reciprocal Lattice Unit Cell

The volume of the reciprocal lattice unit cell is given by \(\mathbf{A} \cdot (\mathbf{B} \times \mathbf{C})\). Calculate \(\mathbf{B} \times \mathbf{C}\) and then its dot product with \(\mathbf{A}\).\[\begin{align*}\mathbf{B} \times \mathbf{C} &= (-\mathbf{k} + \mathbf{j}) \times 2\mathbf{k} \&= 2((-\mathbf{k}) \times \mathbf{k} + \mathbf{j} \times \mathbf{k}) \&= 2(\mathbf{i}).\end{align*}\]Dot product with \(\mathbf{A}\):\[\begin{align*}\mathbf{A} \cdot (\mathbf{B} \times \mathbf{C}) &= (\mathbf{i} - \mathbf{k}) \cdot 2(\mathbf{i}) \&= 2(1 + 0) \&= 2.\end{align*}\]

Step 12: Verify Reciprocal Volume Relationship

The volume calculated for the reciprocal lattice is 2 (from Step 11), which is the reciprocal of the original lattice volume \(\frac{1}{2}\) (from Step 10). Therefore, this confirms that the volumes satisfy the reciprocal relationship: \[2 \times \frac{1}{2} = 1.\]

Key Concepts

Lattice Vectors

Lattice vectors are fundamental building blocks in crystallography and solid-state physics. They are vectors that define the periodic arrangement of points in a crystal, known as a lattice. A three-dimensional lattice is described by three non-coplanar vectors, typically referred to as \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \). Each point in the lattice can be expressed as a linear combination of these vectors with integer coefficients. This means any point \( \mathbf{R} \) in the lattice can be represented as:

• \( \mathbf{R} = n_1 \mathbf{a} + n_2 \mathbf{b} + n_3 \mathbf{c} \)

where \( n_1, n_2, \) and \( n_3 \) are integers. The arrangement forms a repeating pattern, which can define the structure of a crystal. These lattice vectors also help in constructing reciprocal lattices, which are crucial in the study of diffraction patterns.

Cross Product

The cross product is a vector operation important in three-dimensional vector calculus. When you take the cross product of two vectors, you produce a third vector that is orthogonal to both original vectors. The output vector's magnitude depends on the sine of the angle between the original vectors and the magnitudes of both vectors. For vectors \( \mathbf{u} \) and \( \mathbf{v} \), the cross product is denoted by \( \mathbf{u} \times \mathbf{v} \) and can be calculated using the determinant of a matrix:

• \[ \mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ u_1 & u_2 & u_3 \ v_1 & v_2 & v_3 \end{vmatrix} \]

This levi-civita symbol or determinant provides the components of the resulting vector. The cross product is essential in finding reciprocal lattice vectors, as seen where a cross product between two lattice vectors forms part of the reciprocal basis.

Dot Product

The dot product is a scalar product derived from two vectors. It provides a measure of how much one vector extends in the direction of another. For vectors \( \mathbf{u} \) and \( \mathbf{v} \), the dot product is given by \( \mathbf{u} \cdot \mathbf{v} \). This calculates as the sum of the products of their corresponding components:

• \[ \mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + u_3 v_3 \]

Alternatively, the dot product formula can be expressed in terms of the cosine of the angle \( \theta \) between them:

• \[ \mathbf{u} \cdot \mathbf{v} = |\mathbf{u}| |\mathbf{v}| \cos(\theta) \]

This product is particularly useful in determining the volumetric relationships of unit cells in lattices, such as calculating the scalar triple product in determining the volume of a lattice cell.

Vector Calculus

Vector calculus deals with differential and integral calculus applied to vector fields. It is a crucial mathematical tool in physics, particularly when dealing with fields and continuous spatial distributions. Key operations in vector calculus include the gradient, divergence, and curl. These operators help describe how vector fields fluctuate over space. For example, gradients show the rate of change of scalar fields, and divergence indicates the magnitude of a source or sink at a given point in a vector field.In the context of reciprocal lattices, vector calculus is utilized to manipulate and convert spatial relationships, such as determining reciprocal vectors:\( \mathbf{A}, \mathbf{B}, \mathbf{C} \), from real-space lattice vectors:\( \mathbf{a}, \mathbf{b}, \mathbf{c} \). These reciprocal vectors help translate the real-space lattice into reciprocal space, which is necessary when exploring wave phenomena like diffraction. Understanding the logical flow of vectorial operations using cross and dot products is crucial to mastering lattice-related problems.