Problem 120 - Chemistry: The Central Science in SI Units, Global Edition

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 \(154 \mathrm{pm}\) 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 crystal can be found using the Bragg equation \(n\lambda = 2d \sin\theta\). Given a wavelength of 154 pm, an angle of diffraction of 14.22°, and first-order diffraction (n=1), the distance (d) is approximately 313.9 pm.

Step 1: Understand the Bragg Equation

The Bragg equation relates the wavelength of radiation with the angle of diffraction and the distance between planes of atoms in a crystal. It is expressed as: \[n\lambda = 2d \sin\theta\] where \(n\) is the order of diffraction, \(\lambda\) is the wavelength of the radiation, \(d\) is the distance between planes of atoms in the crystal, and \(\theta\) is the angle of diffraction.

Step 2: Given Data

We are given the following information: - Wavelength of radiation (λ) = 154 pm - Angle of diffraction (θ) = 14.22° - Order of diffraction (n) = 1

Step 3: Convert the Angle to Radians

The angle of diffraction is given in degrees, so we need to convert it to radians before using the Bragg equation. To convert degrees to radians, we use the following formula: \[\text{Radians} = \frac{\text{Degrees}}{180} \times \pi\] Plugging in the angle in degrees: \(\theta = \frac{14.22}{180} \times \pi = 0.2481\) radians.

Step 4: Use the Bragg Equation

Now that we have the angle θ in radians, we can apply the Bragg equation to find the distance between planes of atoms (d). First, we need to rewrite the equation for d: \[d = \frac{n\lambda}{2 \sin\theta}\]

Step 5: Plug in the Values and Calculate d

Insert the given values for λ, n, and the calculated value for θ in radians: \(d = \frac{(1)(154 \times 10^{-12})}{2 \sin(0.2481)}\) Now calculate the value of d: \(d = \frac{154 \times 10^{-12}}{2 \times 0.24567} = 313.9 \times 10^{-12} \: \text{meters}\)

Step 6: Convert Meters to Picometers

Since our original wavelength was given in picometers, it's convenient to express our answer in the same units for easier comparison. To convert meters to picometers, simply multiply by \(10^{12}\): \(d = 313.9 \times 10^{-12} \times 10^{12} = 313.9 \: \text{pm}\) The distance between the planes of atoms responsible for diffraction in the crystalline silicon crystal is approximately 313.9 pm.

Key Concepts

Bragg equation

The Bragg equation is a fundamental relation in the study of crystal structures using X-ray diffraction. It is a tool used to determine the spacing between layers of atoms in a crystal. The formula is expressed as:

• \(n\lambda = 2d \sin\theta\)

Here, \(n\) stands for the order of diffraction, which is a whole number. The wavelength \(\lambda\) represents the distance between consecutive peaks of waves, and \(\theta\) is the angle at which these waves are diffracted. Finally, \(d\) symbolizes the distance between planes of atoms within the crystal.
The Bragg equation can be visualized as a reflection equation, similar to how light reflects off surfaces, indicating that X-rays reflect off parallel planes in a crystal at distinct angles. By analyzing these angles, the equation helps in deriving how far apart atomic planes are. This knowledge is essential for understanding the internal structure of the crystal.

wavelength

The wavelength is a key component in use when studying X-ray diffraction and is crucial to the application of the Bragg equation. It refers to the distance between successive peaks (or troughs) in a wave. Wavelength is typically denoted by \(\lambda\) and is measured in units like picometers (pm), nanometers (nm), or meters (m).
In X-ray diffraction, knowing the wavelength of the X-rays is critical since this information feeds into the Bragg equation. Variations in wavelength can affect the calculations of the distance between atomic planes, which influences the interpretation of the crystal's structure. Understanding the concept of wavelength helps learners grasp the physics behind wave interactions, such as diffraction and interference, which are at the heart of techniques like X-ray crystallography.
The higher the frequency of the X-rays, the shorter the wavelength, resulting in greater energy. This powerful characteristic of X-rays allows them to penetrate and provide insights into atomic structures, which are too small to be observed directly with visible light.

angle of diffraction

The angle of diffraction, denoted by \(\theta\), is central to understanding and using the Bragg equation. This angle is the measure at which X-rays hit the surface of a crystal and are scattered. When X-rays encounter a crystalline material, they are diffracted into specific directions based on the crystal's atomic arrangement.
Calculating the exact angle \(\theta\) is a pivotal step, as it affects the determination of the distance \(d\) between layers of atoms in the crystal. Usually given in degrees, this angle needs to be converted into radians for calculations involving trigonometric functions within the Bragg equation.
The significance of the diffraction angle lies in its relation to the constructive interference of waves. Constructive interference occurs when waves combine to form a resultant wave of greater amplitude. The angle \(\theta\) contributes to the conditions necessary for constructive interference, yielding precise values that reveal the internal architecture of a crystal. Understanding this concept helps scientists draw detailed conclusions about material properties based on their crystal structures.

crystal structure

Crystal structure refers to the orderly arrangement of atoms or molecules within a crystal. It is the pattern in which atoms are organized in three-dimensional space. Understanding the crystal structure is essential because it defines the properties and functionalities of the material.
Crystal structures are categorized into different lattice systems, each characterized by unique unit cell parameters like angles and atomic spacings. Knowing the arrangement of atoms aids in the determination of how materials interact with X-rays, leading to patterns that can be analyzed through X-ray diffraction.
The Bragg equation leverages the known geometrical arrangement of atoms in a crystal to calculate the exact dimensions between planes within this structure. By accurately determining these dimensions, researchers can deduce material properties such as density and elasticity, which are vital for applications in various technological fields. Comprehending crystal structures also assists in the development and improvement of pharmaceuticals, metals, semiconductors, and more, by manipulating the atomic arrangement to achieve desired properties.