Question

When $X$ rays of wavelength $0.090\mathrm{nm}$ are diffracted by a metallic crystal, the angle of first-order diffraction $(n=1)$ is measured to be ${15.2}^{\circ }.$ What is the distance (in pm) between the layers of atoms responsible for the diffraction?

Short answer

The distance between the layers is approximately 171.8 pm.

Step-by-step solution

Identify the Formula for Bragg's Law

The problem involves X-ray diffraction, which can be analyzed with Bragg's Law. Bragg's Equation is given by: $$n\lambda =2d\sin (\theta )$$ where $n$ is the order of diffraction, $\lambda$ is the wavelength, $d$ is the distance between layers, and $\theta$ is the angle of diffraction.

Plug in the Given Values

From the problem, we know that $n=1$, $\lambda =0.090\text{ nm}=90\text{ pm}$, and $\theta ={15.2}^{\circ }$. These values will be substituted into Bragg's Law to solve for $d$.

Solve for the Spacing $d$

Rearrange Bragg's equation to solve for $d$: $$d=\frac{n\lambda }{2\sin (\theta )}$$ Substitute the known values: $$d=\frac{1\times 90}{2\sin ({15.2}^{\circ })}$$.

Calculate $\sin (\theta )$

First, find $\sin ({15.2}^{\circ })$ using a calculator: $\sin ({15.2}^{\circ })\approx 0.262$.

Compute the Distance $d$

Substitute $\sin ({15.2}^{\circ })$ into the equation: $$d=\frac{90}{2\times 0.262}\approx \frac{90}{0.524}\approx 171.8\text{ pm}$$.

Conclusion

The distance between the layers of atoms responsible for the diffraction is approximately $171.8\text{ pm}$.

Key concepts

X-ray diffraction

X-ray diffraction is a scientific technique used to study the arrangement of atoms within a crystalline structure. When X-rays hit a crystal, they can be scattered in different directions. This scattering creates a diffraction pattern, which is unique to the structural arrangement of atoms in the crystal. This method is essential in fields such as material science, chemistry, and biology because it helps to determine the atomic and molecular structure of a crystal. Understanding X-ray diffraction allows scientists to unravel the 3D structure of complex molecules like proteins and DNA. In essence, X-ray diffraction acts like a magnifying glass for atoms, making it a critical tool in modern science.

• The process involves the interaction of X-rays with the atomic layers in the crystal.
• The angles and intensities of the diffracted X-rays provide information about the atomic arrangement.
• Analysis of the diffraction pattern helps in defining the dimensions and symmetry of the crystal.

Wavelength measurement

Wavelength is a crucial aspect in the study of diffraction, particularly in X-ray diffraction. It refers to the distance between consecutive peaks of a wave and is typically measured in nanometers (nm) for X-rays.The wavelength of the X-ray beam affects the diffraction pattern. In Bragg's Law, defined by the equation $$n\lambda =2d\sin (\theta )$$, $\lambda$ denotes the wavelength of the incident X-rays. Knowing the exact wavelength is essential for calculating the distance between the atomic layers in a crystal (known as the interplanar spacing $d$).

• Shorter wavelengths provide more detailed information about the crystal structure.
• X-rays have very short wavelengths, typically on the order of a few angstroms (1 angstrom = 0.1 nm).
• Accurate wavelength measurement results in precise determination of atomic distances within a crystal.

Crystalline structure

A crystalline structure refers to the highly ordered arrangement of atoms in a solid material, with a repeating pattern extending in all three spatial dimensions. Each repeating unit, known as the unit cell, builds up the entire crystal by repeating in different directions. Crystalline structures are central to understanding material properties because they affect the mechanical, optical, and electronic behaviors of materials. Studying these structures using techniques like X-ray diffraction gives insight into the fundamental building blocks of materials.

• Crystals differ structurally based on their atomic layout, which can range from cubic to hexagonal arrangements.
• The symmetry and orientation of atoms in a crystal impact the diffraction pattern observed.
• Understanding the crystalline structure aids in developing materials with specific desired properties.

Atomic layer spacing

Atomic layer spacing, often referred to as interplanar spacing, is the distance between parallel planes of atoms in a crystalline material. This spacing is critical for understanding how materials diffract X-rays. Bragg's Law, often used in X-ray diffraction studies, includes this spacing as a key variable. By knowing the X-ray wavelength and the angle at which X-rays are diffracted, scientists can calculate the interplanar spacing using Bragg's equation:$$d=\frac{n\lambda }{2\sin (\theta )}$$where $d$ is the interplanar spacing, $\lambda$ is the wavelength, and $\theta$ is the diffraction angle.

• Interplanar spacing helps in determining the structure and symmetry of the crystal.
• The precise measurement of this spacing is crucial for material characterization.
• Understanding atomic layer spacing is essential in designing advanced materials and technologies.