Question

In their study of X-ray diffraction, William and Lawrence Bragg determined that the relationship among the wavelength of the radiation \((\lambda)\), the angle at which the radiation is diffracted \((\theta)\), and the distance between the layers of atoms in the crystal that cause the diffraction \((d)\) is given by \(n \lambda=2 d \sin \theta\). (a) X-rays from a copper X-ray tube that have a wavelength of \(1.54 \AA\) are diffracted at an angle of \(14.22\) degrees by crystalline silicon. Using the Bragg equation, calculate the interplanar spacing in the crystal, assuming \(n=1\) (first-order diffraction). (b) Repeat the calculation of part (a) but for the \(n=2\) case (second-order diffraction).

Short answer

The interplanar spacing in the crystal is: (a) \(3.14 \AA\) for the first-order diffraction \((n=1)\), and (b) \(6.27 \AA\) for the second-order diffraction \((n=2)\).

Step-by-step solution

Understand the Bragg equation

The given equation is called Bragg's law and relates the wavelength \((λ)\), angle of diffraction \((θ)\), and distance between layers of atoms \((d)\) to the order of diffraction \((n)\). \(n λ = 2 d \sin θ\) First, we will solve for the interplanar spacing \((d)\) with first-order diffraction \((n=1)\), and then we will repeat the calculation for the second-order diffraction \((n=2)\).

Calculate interplanar spacing for the first-order diffraction

Using the given values, calculate the interplanar spacing with first-order diffraction \((n=1)\). Given: \(λ = 1.54 \AA\), \(θ = 14.22 \degree\), \(n = 1\). First, we need to convert the angle of diffraction from degrees to radians: \[ θ = 14.22 \degree \times \frac{\pi}{180} = 0.248 \text{ radians} \] Now, we can substitute values into Bragg's equation to find the interplanar spacing \((d)\): \[ 1\times 1.54 \AA = 2 d \sin 0.248 \] Solve for \(d\): \[ d = \frac{1.54 \AA}{2\sin 0.248} = 3.14 \AA \] So, the interplanar spacing for the first-order diffraction is \(d = 3.14 \AA\).

Calculate interplanar spacing for the second-order diffraction

Using the given values, calculate the interplanar spacing with second-order diffraction \((n=2)\). Given: \(λ = 1.54 \AA\), \(θ = 14.22 \degree\), \(n = 2\). We already have the angle of diffraction in radians, so we can substitute the values into Bragg's equation to find the interplanar spacing \((d)\): \[ 2\times 1.54 \AA = 2 d \sin 0.248 \] Solve for \(d\): \[ d = \frac{2 \times 1.54 \AA}{2\sin 0.248} = 6.27 \AA \] So, the interplanar spacing for the second-order diffraction is \(d = 6.27 \AA\). #Conclusion# The interplanar spacing in the crystal is: (a) \(3.14 \AA\) for the first-order diffraction \((n=1)\). (b) \(6.27 \AA\) for the second-order diffraction \((n=2)\).

Key concepts

X-ray Diffraction

Understanding X-ray diffraction is essential when it comes to analyzing the structure of crystalline materials. It's a powerful technique that involves directing X-rays at a material and observing how they scatter off the atoms within it. The patterns of scattered X-rays can be used to determine the arrangement of atoms in a crystal, revealing the crystal's inner structure.
One of the underlying principles of X-ray diffraction is that when the X-rays encounter the regular array of atoms in a crystal, they can constructively and destructively interfere with one another, producing a diffraction pattern. The areas of constructive interference, known as diffraction peaks, are of particular interest because they contain information about the crystal's geometry.

Interplanar Spacing

Interplanar spacing, denoted by d, refers to the distance between adjacent planes of atoms in a crystal. These planes are imaginary but represent the regular, repeating units that make up the crystal lattice. Interplanar spacing is a critical parameter that can be calculated using Bragg's law, given the wavelength of the X-rays and the angle at which they are diffracted.
Accurate determination of interplanar spacing is crucial since it directly relates to the size and shape of the unit cell of the crystal, providing insight into the material's physical properties and potential applications. For instance, in the provided exercise, the calculation of interplanar spacing for two different orders of diffraction demonstrates how the same crystal can produce multiple diffraction patterns at various scales.

Crystallography

Crystallography is the scientific study of crystals and the arrangement of atoms within solids. The goal of crystallography is to acquire a comprehensive understanding of the structure of materials at the atomic or molecular level. Techniques like X-ray diffraction play a pivotal role in crystallography because they allow researchers to peer into the crystal lattice where direct observation is not possible.
Through the interpretation of diffraction patterns, crystallographers can infer the positions of atoms, the bonds between them, and the symmetry of the crystal structure. These insights contribute to fields as diverse as materials science, chemistry, biology, and physics, aiding in the development of new materials and the study of proteins and other complex molecules.

Diffraction Angle

The diffraction angle, often represented as θ in Bragg's law, is the angle at which X-rays are scattered from the crystal planes. It is a crucial variable because it directly influences the resulting diffraction pattern. The position and intensity of the diffraction peaks are highly sensitive to changes in this angle, which is why it has to be measured precisely in experiments.
Diffraction angles help scientists in distinguishing between different sets of crystal planes within the crystal, as each set will diffract X-rays at unique angles. In the context of the provided exercise, the diffraction angle is used, along with the wavelength of the X-rays and the order of diffraction, to calculate the interplanar spacing. It is also worth noting that diffraction angles are commonly converted from degrees to radians in calculations, as radians are the standard unit of angular measure in physics.