Question

Calculate the interplanar spacings (in picometers) that correspond to diffracted beams of \(X\) rays at \(\theta=20.0^{\circ}\), \(27.4^{\circ},\) and \(35.8^{\circ},\) if the \(X\) rays have a wavelength of \(141 \mathrm{pm}\). Assume that \(n=1\).

Short answer

The interplanar spacings for the angles 20.0^\circ, 27.4^\circ, and 35.8^\circ are approximately 207.24 pm, 155.53 pm, and 124.49 pm respectively.

Step-by-step solution

Understand Bragg's Law

Bragg's Law describes the relationship between the angle of incidence and the wavelength of X-rays that result in constructive interference from a crystal lattice. The law is given by the equation: \( n\lambda = 2d\sin(\theta) \), where \( n \) is the order of the reflection, \( \lambda \) is the wavelength of the X-rays, \( d \) is the interplanar spacing, and \( \theta \) is the angle of incidence.

Calculate the interplanar spacing for \( \theta = 20.0^\circ \)

Substitute the known values into Bragg's Law for \( \theta = 20.0^\circ \): \( 1 \cdot 141 \text{pm} = 2d\sin(20.0^\circ) \).Then, solve for \( d \) to find the interplanar spacing:\( d = \frac{141\text{pm}}{2\sin(20.0^\circ)} \).

Calculate the interplanar spacing for \( \theta = 27.4^\circ \)

Using the same method, substitute \( \theta = 27.4^\circ \) into Bragg's Law:\( d = \frac{141\text{pm}}{2\sin(27.4^\circ)} \).

Calculate the interplanar spacing for \( \theta = 35.8^\circ \)

Likewise, substitute \( \theta = 35.8^\circ \) into Bragg's Law:\( d = \frac{141\text{pm}}{2\sin(35.8^\circ)} \).

Calculate actual interplanar spacings

Now we perform the calculations for each \( \theta \) value to find the corresponding interplanar spacings: For \( \theta = 20.0^\circ \):\( d = \frac{141\text{pm}}{2\sin(20.0^\circ)} \approx 207.24 \text{pm} \).For \( \theta = 27.4^\circ \):\( d = \frac{141\text{pm}}{2\sin(27.4^\circ)} \approx 155.53 \text{pm} \).For \( \theta = 35.8^\circ \):\( d = \frac{141\text{pm}}{2\sin(35.8^\circ)} \approx 124.49 \text{pm} \).

Key concepts

Interplanar Spacing

Interplanar spacing refers to the distance between parallel planes of atoms inside a crystal structure. It's essential for understanding how X-rays interact with crystals, as these planes act like mirrors reflecting the rays. In the context of Bragg's Law, the interplanar spacing is critical for determining the angles at which X-rays are diffracted by the crystal.

To visualize interplanar spacing, imagine a stack of equally spaced sheets of paper. Each sheet represents an atomic plane. The space between the sheets represents the interplanar spacing in a crystal. Bragg's Law provides us a way to calculate this spacing using the X-ray diffracted angles and the wavelength of the rays.

By knowing this spacing, we can learn a lot about the crystal's structure, including the size and shape of its unit cells, and the arrangement of its atoms.

X-Ray Diffraction

X-ray diffraction (XRD) is a powerful technique used in materials science to study the structure of crystalline materials. When X-rays hit a crystal, they are scattered by atoms within the crystal lattice. If the conditions described by Bragg's Law are met, the scattered waves undergo constructive interference and produce a diffraction pattern.

The diffraction pattern is characteristic of the atomic arrangement within the crystal and can be recorded on a detector. By analyzing this pattern, scientists can determine the crystal structure, including the positions of atoms and any defects in the crystal. XRD analysis is a cornerstone in the field of crystallography and is essential for material identification and characterization.

Constructive Interference

Constructive interference is a key concept in the study of wave phenomena, which also applies to X-ray waves in diffraction studies. It occurs when multiple waves overlap and combine to produce a wave of greater amplitude.

What Causes Constructive Interference?

In X-ray diffraction, constructive interference happens when the path difference between the waves reflected from different atomic planes is an integer multiple of the wavelength. This condition, governed by Bragg's Law, results in an intensified beam, or peak, in the diffraction pattern. Constructive interference allows us to detect the presence of certain planes within the crystal and gather information about the crystal's internal structure.

Crystal Lattice

A crystal lattice is a three-dimensional, periodic arrangement of atoms, ions, or molecules in a crystal. Each point in the lattice, known as a lattice point, represents the position of an atom or a group of atoms that has the same environment throughout the entire crystal.

Crystals have a unique and well-defined lattice structure, which is reflected in the symmetry and sharp edges of the natural mineral forms. Understanding the crystal lattice is crucial, as it determines the crystal's physical properties, such as its thermal and electrical conductivity, optical properties, and strength. X-ray diffraction is the primary method to investigate crystal lattices, providing us with information about the atomic arrangement and allowing the identification of new materials and the synthesis of novel compounds with desired properties.