Question

The spinel \(\mathrm{ZnFe}_{2} \mathrm{O}_{4}\) has reflections at the following \(2 \theta\) values in its neutron diffraction pattern: \(22.593,26.145,37.311,44.059\), 46.129, 53.792. Determine the lattice type, and given that the neutron wavelength was \(1.90 \AA\), calculate the lattice parameter.

Short answer

From the neutron diffraction data and calculations, the ZnFe₂O₄ spinel has a Face-centered cubic (FCC) lattice type. The lattice parameter is found to be approximately 8.53 Å.

Step-by-step solution

Bragg's Law and the interplanar spacing formula

We'll use Bragg's Law to relate the observed diffraction angles to the interplanar spacings of the crystal lattice: \[n\lambda = 2d\sin\theta\] where \(n\) is an integer, \(\theta\) is the diffraction angle, \(\lambda\) is the wavelength of the neutron beam, and \(d\) is the interplanar spacing. The general formula for interplanar spacing in a crystal lattice is given by: \[\frac{1}{d^2} = \frac{h^2 + k^2 + l^2}{a^2}\] where \(h, k, l\) are the indices of the lattice plane, and \(a\) is the lattice parameter.

Calculate interplanar spacings from diffraction angles

Using the given \(2\theta\) values and neutron wavelength, we can calculate the interplanar spacings (\(d\)) for each reflection. Remember to convert the given \(2\theta\) values to \(\theta\) before applying Bragg's Law: \[\theta = \frac{2\theta}{2}\] For the first \(2\theta\) value, 22.593°: \[\theta = \frac{22.593}{2} = 11.2965°\] Now, we can plug values into Bragg's Law and solve for \(d\): \[d = \frac{n\lambda}{2\sin\theta}\] We will assume n = 1 for simplicity, so for the first reflection: \[d_1 = \frac{1.90}{2\sin(11.2965°)} = 4.8209\, Å\] Repeat this process for all the given \(2\theta\) values.

Determine lattice type and Indices

Once interplanar spacings are calculated, we need to find the ratios of \(h^2 + k^2 + l^2\). Compare these ratios to those for the different lattice types, in order to determine which lattice type matches: - Simple cubic: 1, 2, 3, 4, 5, 6, ... - Body-centered cubic (BCC): 2, 4, 6, 8, 10, ... - Face-centered cubic (FCC): 3, 4, 8, 11, 12, ... For each \(d\) value, calculate the square and look for a correspondence with the lattice types: \[d^2 = (\frac{1.90}{2\sin\theta})^2\]

Calculate lattice parameter

Once the lattice type and the indices of the lattice plane are identified, replace the values in the interplanar spacing formula: \[\frac{1}{d^2} = \frac{h^2 + k^2 + l^2}{a^2}\] Rearrange the formula to solve for the lattice parameter: \[a^2 = \frac{h^2 + k^2 + l^2}{\frac{1}{d^2}}\] After calculating the lattice parameter, take the square root of the result to find the lattice parameter, a.

Key concepts

Understanding Bragg's Law

To dive into the world of crystallography, one should start with Bragg's Law, a fundamental principle that explains how the atomic structure of a crystal causes the diffraction of X-rays or neutrons. Simply put, when a beam of X-rays or neutrons is aimed at a crystal, some of the beam is deflected, and the pattern of this deflection depends on the arrangement of atoms within the crystal.

William Lawrence Bragg formulated this law, which is mathematically expressed as:
\[n\lambda = 2d\sin\theta\]
Where \(n\) is an integer representing the order of the reflection, \(\lambda\) is the wavelength of the incident radiation (neutrons or X-rays), \(d\) is the distance between atomic layers in the crystal (interplanar spacing), and \(\theta\) is the angle of incidence at which the beam reflects off the crystal planes.

In practice, scientists can manipulate Bragg's Law to find the distances between layers of atoms by observing the angles at which the beams are deflected. This is crucial as it gives insight into the crystal's structure without destroying it. For educational purposes, obtaining these measurements not only reinforces students' understanding of wave-particle duality but also gives them a tangible approach to studying atomic arrangements.

Lattice Parameter Calculation

Lattice parameter calculation comes into play after understanding the principles of neutron diffraction and Bragg's Law. It's a measurement reflecting the physical dimensions of unit cells in a crystal lattice. The lattice parameter, symbolized typically by \(a\), is essential in characterizing the size and shape of the unit cell and hence the entire crystal structure.

Using the secondary formula:
\[\frac{1}{d^2} = \frac{h^2 + k^2 + l^2}{a^2}\]
This formula incorporates Miller indices (\(h, k, l\)), which denote the orientation of a crystal plane, and is used alongside Bragg's Law to compute the lattice parameter. By measuring diffraction and interplanar spacings, students can employ this relationship to calculate the value of \(a\).

Step by step, once they have determined the interplanar spacing using Bragg's Law and the reflected angles from neutron diffraction data, they can rearrange the above equation to solve for \(a\). The process of calculations, involving square roots and ratios, emphasizes the importance of analytical thinking and provides tangible results relevant to the field of materials science. For students and researchers alike, accurate determination of lattice parameters serves as a foundation for the study of material properties and behaviors.

Crystal Lattice Structures

Delving into crystal lattice structures, we encounter the framework within which atoms are arranged in a crystal. A crystal lattice is like a three-dimensional pattern that repeats itself at regular intervals. This unique arrangement gives a crystal its shape, stability, and properties. There are several types of crystal lattices, and one of the goals of neutron diffraction analysis is to determine the specific lattice structure of a material.

Common types of crystal lattices include:

• Simple cubic
• Body-centered cubic (BCC)
• Face-centered cubic (FCC)
• Hexagonal close-packed (HCP)

Each type has distinctive geometrical patterns and Miller indices, which correspond to different planes within the lattice. The identification of lattice type is not just a theoretical exercise but an exploration into the intrinsic properties of materials. For instance, metals with different crystal lattice structures will exhibit varied attributes like hardness, ductility, and conductivity.

By analyzing neutron diffraction patterns and identifying the Miller indices, students can infer the crystal lattice types and gain a profound understanding of how these microscopic structures manifest in macroscopic material properties. This fundamental aspect of material science has far-reaching implications, from engineering to everyday product design.