Question

The second-order diffraction \((n=2)\) for a gold crystal is at an angle of \(22.20^{\circ}\) for \(X\) rays of 154 pm. What is the spacing between these crystal planes?

Short answer

The spacing between the crystal planes is approximately 402.70 pm.

Step-by-step solution

Write down Bragg's Law

Bragg's Law is given by the formula: \[n\lambda = 2d\sin{\theta}\] Where: - \(n\) is the order of diffraction, - \(\lambda\) is the wavelength of the X-rays, - \(d\) is the spacing between the crystal planes, and - \(\theta\) is the angle of diffraction.

Plug in the given values into Bragg's Law

We have been given the following values: - \(n = 2\) (second-order diffraction), - \(\lambda = 154 \, \text{pm}\) (wavelength of the X-rays), and - \(\theta = 22.20^\circ\) (angle of diffraction). Now substitute these values in Bragg's Law: \(2(154 \, \text{pm}) = 2d \sin{22.20^\circ}\)

Solve the equation for the spacing between the crystal planes (d)

To find the spacing between the crystal planes, \(d\), we now just need to solve the equation from Step 2: \(308 \, \text{pm} = 2d \sin{22.20^\circ}\) Next, divide both sides of the equation by \(2\sin{22.20^\circ}\): \(d = \frac{308 \, \text{pm}}{2 \sin{22.20^\circ}}\) Finally, compute the value of \(d\): \(d \approx 402.70 \, \text{pm}\) The spacing between the crystal planes is approximately 402.70 pm.

Key concepts

Diffraction

Diffraction is a fascinating concept in physics that explains the behavior of waves as they encounter obstacles or pass through narrow openings. This phenomenon is especially significant when studying light and other types of waves, like X-rays. When waves encounter an object or a gap, they tend to bend around it, leading to interference patterns. These patterns can reveal a wealth of information about the object's structure.

In the context of Bragg's Law and crystallography, diffraction plays a crucial role. As X-ray beams hit a crystalline material, the waves are diffracted by the crystal lattice. This creates an interference pattern that can be analyzed to determine various properties of the crystal, such as the spacing between its planes. Understanding diffraction is key to unlocking the secrets hidden within the microscopic world of crystals.

Crystal Planes

Crystal planes are integral components of a crystalline substance. Imagine slicing the crystal into smooth, evenly spaced layers, much like cutting through a block of steel into thin sheets. These layers are referred to as crystal planes, and they define the distinctive structure of a crystal lattice.

In crystallography, the spacing between these planes, represented as "d" in Bragg's Law, provides vital insights into the material's internal structure. By understanding the arrangement and spacing of these planes, scientists can determine essential properties like crystal symmetry, size of unit cells, and even the type of chemical bonds present in the material.

• The arrangement of atoms within these planes dictates many physical properties of the crystal.
• Bragg's Law allows for the calculation of the spacing between these planes, which is crucial for material analysis.

Essentially, crystal planes are like the fingerprints of a crystal, offering clues about its structure and behavior.

X-ray Wavelength

X-rays are a form of electromagnetic radiation, similar to visible light but with much higher energy and shorter wavelengths. These rays are particularly useful in studying crystal structures due to their ability to penetrate deeply and scatter off the densely packed atoms within the crystal lattice.

A critical factor in determining a crystal's structure using X-rays is understanding the wavelength, represented as "\(\lambda\)" in Bragg's Law. The wavelength must be on the order of the spacing between crystal planes to produce meaningful diffraction patterns. When X-rays of a known wavelength are directed at a crystal, the angles at which they are scattered can give precise measurements of the distances between the planes.

• The wavelength of X-rays used is often in picometers (pm), a unit suitable for atomic-scale dimensions.
• Adjustments in wavelength can allow for the analysis of various crystal structures, revealing details not visible with other forms of radiation.

Therefore, the proper use of X-ray wavelengths is a powerful tool in the study and understanding of crystalline structures.