Question

Diffraction of \(X\) rays with \(\lambda=154.2 \mathrm{pm}\) occurred at an angle \(\theta=22.5^{\circ}\) from a metal surface. What is the spacing (in \(\mathrm{pm}\) ) between the layers of atoms that diffracted the X rays?

Short answer

The spacing between the layers is approximately 201.5 pm.

Step-by-step solution

Understand the Bragg's Law Formula

Bragg's Law is used to find the spacing between crystal layers that cause diffraction. The law is given by the equation \( n\lambda = 2d\sin\theta \), where \( n \) is the order of reflection (usually 1 for first order), \( \lambda \) is the wavelength of the incident wave, \( d \) is the distance between layers, and \( \theta \) is the angle of incidence.

Organize the Given Information

From the problem, we know that \( \lambda = 154.2 \, \text{pm} \) and \( \theta = 22.5^\circ \). We will assume first order reflection so \( n = 1 \). We need to find \( d \), the spacing between layers.

Solve for the Spacing \( d \)

Rearrange Bragg's Law to solve for \( d \): \( d = \frac{n\lambda}{2\sin\theta} \). Substitute the known values into the equation: \( d = \frac{1 \times 154.2 \, \text{pm}}{2 \sin 22.5^\circ} \).

Calculate \( d \) Using the Formula

Calculate \( \sin 22.5^\circ \) which approximately equals 0.3827. Now substitute back into the formula: \( d = \frac{154.2}{2 \times 0.3827} \approx \frac{154.2}{0.7654} \approx 201.5 \, \text{pm} \).

Verify Your Calculation

Verify the division results: divide 154.2 by 0.7654 again to ensure the calculated spacing \( d \) is consistent. The spacing is approximately 201.5 pm.

Key concepts

X-ray diffraction

X-ray diffraction is a powerful technique used to study the structure of crystalline materials. It relies on the way X-rays scatter when they encounter a crystal lattice, which is a regular arrangement of atoms in solid materials.
When these X-rays strike the crystal, they are diffracted in specific directions depending on the arrangement of atoms within the crystal. This diffraction pattern provides a fingerprint of the crystal's structure.
Bragg's Law is fundamental in X-ray diffraction, allowing scientists to decipher the spacing between atomic layers in crystals. The law states that diffraction occurs when all the conditions specified by the formula are met. This formula is:

• \( n\lambda = 2d\sin\theta \)

Here, \( n \) is the order of diffraction (commonly equal to one, for first-order diffraction). \( \lambda \) represents the wavelength of the X-ray, \( d \) is the spacing between layers of atoms, and \( \theta \) is the angle of incidence. As X-rays pass through a crystal, they produce a diffraction pattern that reveals the geometric arrangement of the crystal structure.
Understanding how to harness this diffraction pattern is key in fields like materials science, chemistry, and physics.

atomic layer spacing

The term "atomic layer spacing" refers to the distance between layers of atoms within a crystalline structure. When performing X-ray diffraction analysis, scientists often aim to measure this atomic layer spacing.
Determining the distance between these layers is crucial for understanding how atoms are packed together in the crystal, which in turn affects the material's properties.
Using Bragg's Law, we can calculate this spacing using the wavelength of the X-rays and the angle at which diffraction occurs. The formula restructures as:

• \( d = \frac{n\lambda}{2\sin\theta} \)

This equation helps to measure the spacing \( d \) without directly observing the atoms. It allows for an indirect investigation of the atomic structure.
By using the known values of \( \lambda \) and \( \theta \), one can substitute back into the equation to find \( d \). This is critical for unlocking the structural properties of materials.

crystal structure analysis

Crystal structure analysis involves the use of X-ray diffraction to understand how atoms are arranged within a crystal. By analyzing the diffraction pattern, scientists can determine the distances between atomic planes and the angles at which these planes intersect.
This structural mapping is essential for several fields such as mineralogy, metallurgy, and pharmaceuticals.
Upon obtaining a diffraction pattern, scientists can interpret the arrangement of atoms, identify the type of crystal lattice, and conclude the material's crystallography. These findings impact the understanding of a material's mechanical, electronic, and thermal properties, making crystal structure analysis a cornerstone in material science.
In a lab setting, researchers can prepare samples and conduct experiments to gather diffraction data. Using this data, computer software can help visualize and simulate the atomic structure. In this way, X-ray diffraction aids in crafting new materials and improving existing ones, by allowing for a precise understanding of atomic arrangements.