Question

\(\beta\)-Tungsten crystallizes with a body-centred cubic structure. What \(h, k, l\) values would you expect the first ten reflections to have?

Short answer

The first 10 possible reflections are (h, k, l) = (1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 1, 0), (1, 0, 1), (0, 1, 1), (1, 1, 1), (2, 0, 0), (0, 2, 0), (0, 0, 2).

Step-by-step solution

Understand the BCC Reflection Rule

In a body-centered cubic (BCC) unit cell, a reflection is possible only when the sum of the Miller indices (h, k, l) is even. This is because, in a BCC structure, the scattered waves from the atoms at the cell corners and the atom at the body center destructively interfere unless h+k+l is even.

List down possible reflections

List down the reflections based on the BCC reflection rule starting in increasing order for sum of Miller indices (h, k, l). Reflections with equivalent Miller indices should be avoided, since they are essentially the same reflection. For instance, we consider \((1, 0, 0)\) and not \((0, 1, 0)\) or \((0, 0, 1)\).

Determine the first 10 possible reflections

The first 10 possible reflections are (h, k, l) = (1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 1, 0), (1, 0, 1), (0, 1, 1), (1, 1, 1), (2, 0, 0), (0, 2, 0), (0, 0, 2).

Key concepts

Miller Indices

Miller indices are a symbolic vector representation used in crystallography to describe the orientation of planes within a crystal lattice. They are denoted by three integers \(h, k, l\), known as the Miller indices, and are derived from the reciprocals of the fractional intercepts that the plane makes with the crystal’s axes.

In the context of the \(\beta\)-Tungsten exercise you came across, Miller indices simplify the task of identifying and predicting the pattern of planes that create reflections within the crystalline structure. When dealing with crystal planes, it’s crucial to remember that equivalent planes have the same atomic arrangements. Thus, reflections from these planes have an identical intensity, and it’s sufficient to consider only one representation to understand the reflection pattern.

Using the BCC reflection rule for \(\beta\)-Tungsten, it becomes straightforward to list the first ten reflections. By ensuring that the sum of \(h, k, l\) is even to satisfy constructive interference, one can quickly identify the permissible reflections and, consequently, the significant planes within the crystal structure.

Body-Centered Cubic Structure

A body-centered cubic (BCC) structure is one of the several atomic arrangements in crystalline materials. In BCC, the unit cell consists of atoms at each of the eight corners of a cube and a single atom at the very center of the cube. This atom at the center effectively interacts with the atoms at the corners, resulting in a distinctive pattern of atomic space filling.

The BCC configuration has a significant impact on the material’s properties, such as its density and the manner in which it diffracts X-rays. Understanding the layout of atoms is essential when determining the crystal planes that contribute to diffraction. In the case of \(\beta\)-Tungsten, knowing that it has a BCC structure allows us to predict its diffraction pattern by applying the BCC reflection rule.

For \(\beta\)-Tungsten, proper identification of potential crystal planes is facilitated by following the BCC reflection rule. This rule becomes a critical factor in predicting which planes would generate the reflections that one might observe in a diffraction experiment.

Destructive Interference in Crystallography

Destructive interference is a phenomenon in crystallography where waves scattered by the atoms within a crystal cancel each other out. This occurs when the difference in path length between waves is equal to an odd number of half wavelengths. In contrast, constructive interference, where waves enhance each other, takes place when the path length difference is a multiple of the full wavelength.

In BCC crystals like \(\beta\)-Tungsten, the atoms at the corners and the center can cause destructive interference unless the Miller indices satisfy certain conditions. For \(\beta\)-Tungsten, it requires that the sum of Miller indices \(h, k, l\) be even, as indicated in the provided exercise solution. This specificity ensures that the waves from all parts of the unit cell constructively interfere, resulting in observable reflections.

Conscious application of the knowledge about destructive interference in the analysis of diffraction patterns leads to a better understanding of the material's internal structure, which can elucidate its physical properties and behaviors. For students studying materials science or crystallography, grasping the concepts of interference is fundamental to interpreting X-ray diffraction patterns and, by extension, the atomic arrangements within crystals.