Problem 106 - Chemistry The Central Science

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 planes of atoms in the crystal that cause the diffraction \((d)\) is given by \(n \lambda=2 d \sin \theta .\) 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 distance between the planes of atoms responsible for diffraction in this crystal, assuming \(n=1\) (first-order diffraction).

Step-by-Step Solution

Answer

The distance between the planes of atoms responsible for diffraction in the crystalline silicon can be calculated using the given values and the Bragg equation. First, convert the angle (14.22 degrees) to radians (approx. 0.248 rad). Then, plug the angle, wavelength (1.54 Å), and first-order diffraction (n=1) into the equation \(n \lambda=2 d \sin \theta \). Solving for the distance (d), we find that it is approximately \(1.91 \AA\).

Step 1: Write down the given values

First, let's write down the given information: - Wavelength of the X-rays \((\lambda) = 1.54 \AA\) (angstroms) - Angle of diffraction \((\theta) = 14.22\) degrees - Bragg equation: \(n \lambda=2 d \sin \theta \) - Order of diffraction \((n) = 1\)

Step 2: Convert angle from degrees to radians

We need to work with angles in radians. To convert the angle from degrees to radians, use the following formula: \[\theta_{rad} = \theta_{deg} \cdot \frac{\pi}{180}\] For our given angle \(\theta\), the conversion is: \[\theta_{rad} = 14.22 \cdot \frac{\pi}{180} \approx 0.248\text{ rad}\]

Step 3: Plug the given values into the Bragg equation

Now, we plug the given values into the Bragg equation and solve for \(d\): \[n \lambda = 2 d \sin \theta\] Since \(n = 1\), the equation simplifies to: \[\lambda = 2d \sin \theta\] Now, plug in the values for \(\lambda\) and \(\theta_{rad}\): \[1.54\AA = 2d \sin 0.248\]

Step 4: Solve for the distance between planes of atoms (d)

Now, we just need to solve for \(d\): \[ d = \frac{1.54\AA}{2\sin 0.248}\] \[ d \approx 1.91\AA\] So, the distance between the planes of atoms responsible for diffraction in this crystal is approximately \(1.91 \AA\).

Key Concepts

Bragg's Law

Bragg's law is a fundamental principle in the field of X-ray diffraction. It describes how X-rays are scattered by the atomic planes in a crystal structure, producing a diffraction pattern. This pattern can be used to determine important structural information about the crystal. The law is given by the equation:

\[ n \lambda = 2d \sin \theta \]

Where:

• \( n \) is the order of diffraction, an integer.
• \( \lambda \) is the wavelength of the incident X-ray beam.
• \( d \) is the distance between the atomic planes.
• \( \theta \) is the angle between the incident X-ray and the scattering plane.

Understanding Bragg's law allows scientists to interpret the diffraction patterns and deduce the crystalline structure of materials. It is widely used in material science, chemistry, and physics.

Wavelength

The concept of wavelength is critical when dealing with X-rays, especially in the context of Bragg's law. Wavelength is the distance between two consecutive peaks or troughs of a wave. In the case of X-rays, it is usually measured in angstroms (\( \AA \)), where 1 angstrom equals \( 10^{-10} \) meters.

The wavelength of the X-ray must be comparable to the spacing between the planes of atoms in the crystal for diffraction to occur. This is because the diffraction peaks are a result of constructive interference, which happens when the path difference between waves reflecting off different planes equals an integer multiple of the wavelength.

Crystalline Structure

Crystalline structure refers to the ordered arrangement of atoms in a material. This structure is characterized by its repeating pattern, which forms a lattice throughout the crystal. The regularity and symmetry of this arrangement are what allow a crystal to diffract X-rays in a predictable pattern.

Different types of crystals have different lattice structures, such as face-centered cubic or body-centered cubic, which affect how X-rays are diffracted. By analyzing the diffraction pattern using Bragg's law, scientists can determine the type of lattice and gain insight into the bonding and properties of the material.

Atomic Planes

Atomic planes refer to the flat, two-dimensional arrays of atoms within a crystal. They act like surfaces that reflect X-ray beams in a diffraction experiment. The spacing between these planes (denoted as \( d \) in Bragg's law) is crucial in determining the diffraction pattern and ultimately the crystalline structure.

In an X-ray diffraction experiment, the angle \( \theta \) at which the X-rays are scattered relates directly to these atomic planes, and knowing \( \theta \) along with the wavelength allows for the calculation of \( d \). Understanding atomic planes is essential for studying and utilizing the properties of crystals in scientific applications.