Question

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

Short answer

The spacing between the crystal planes in the gold crystal can be found using Bragg's Law, \( n\lambda = 2d\sin \theta \). Plugging in the given values, we get \( d = \frac{2 \cdot 154\, \mathrm{pm}}{2\sin(22.20^{\circ})} \approx 200.78\, \mathrm{pm} \). Thus, the spacing between the crystal planes is approximately \(200.78\, \mathrm{pm}\).

Step-by-step solution

Understand Bragg's Law

Bragg's Law is given by \(n\lambda = 2d\sin \theta\), where: - \(n\) is the order of diffraction (in this case, \(n=2\)) - \(\lambda\) is the wavelength of the incident rays (in this case, \(154 \, \mathrm{pm}\)) - \(d\) is the spacing between the crystal planes (the quantity we want to find) - \(\theta\) is the angle of diffraction (in this case, \(22.20^{\circ}\))

Plug in the given values

We will plug in the given values of \(n\), \(\lambda\), and \(\theta\) into Bragg's Law equation: \[2\cdot 154 \, \mathrm{pm} = 2d\sin (22.20^{\circ})\]

Solve for the spacing\(d\)

Now we need to solve for the spacing between the crystal planes, \(d\). To do this, we can divide both sides of the equation by \(2\sin(22.20^{\circ})\). \[d = \frac{2\cdot 154\, \mathrm{pm}}{2\sin(22.20^{\circ})}\]

Calculate the final answer

Finally, we can plug in the values and calculate the spacing between the crystal planes, \(d\): \[d = \frac{2 \cdot 154\, \mathrm{pm}}{2\sin(22.20^{\circ})} \approx 200.78\, \mathrm{pm}\] The spacing between the crystal planes in the gold crystal is approximately \(200.78\, \mathrm{pm}\).

Key concepts

X-ray diffraction

X-ray diffraction is a powerful technique used to study the atomic and molecular structure of crystals. When X-rays are directed onto a crystal, they interact with the electrons and are scattered in many directions. By measuring the angles and intensities of these scattered rays, scientists can deduce information about the electron structure, and therefore, the arrangement of atoms within the crystal.
This process is important because X-rays have wavelengths that are on the same order as the distances between atoms in a crystal, allowing them to effectively "see" into the crystal structure. The scattered X-rays can show constructive or destructive interference, which is dependent on the arrangement of atoms. The specific angles that cause constructive interference give rise to a pattern called an X-ray diffraction pattern. These patterns are unique to each crystal structure and can be used to identify materials or to analyze their physical and chemical properties.

Crystal plane spacing

Crystal plane spacing, denoted as \(d\), refers to the distance between parallel planes of atoms within a crystal. This measurement is crucial in understanding the crystal's lattice structure and is a key parameter in determining the properties of a material.
Using Bragg's Law, \(n\lambda = 2d\sin \theta\), we can find the crystal plane spacing by rearranging the equation to solve for \(d\). This is done by using known values of the wavelength \(\lambda\), the order of diffraction \(n\), and the angle of diffraction \(\theta\).

• \(n\): Integer representing the order of the diffracted peak, showing how many wavelengths fit into a path difference.
• \(\lambda\): The wavelength of the X-rays used in the diffraction.
• \(\theta\): The angle at which the diffracted peak is observed.

By substituting these values into the equation, the spacing \(d\) can be calculated. In materials science, knowing \(d\) helps in identifying the type of crystal and predicting its mechanical and optical properties.

Second-order diffraction

In X-ray diffraction, a diffracted beam can occur at various orders, indicated by the integer \(n\) in Bragg's Law. Second-order diffraction, or \(n=2\), occurs when the path difference between the x-rays reflected from successive crystal planes is equal to two wavelengths.
This means that the X-ray beam can reflect off additional planes of atoms within the crystal, resulting in a different angle of diffraction compared to first-order diffraction \((n=1)\).

• Multiple orders of diffraction provide more data points for interpreting the X-ray diffraction pattern, leading to more accurate determination of the crystal structure.
• Each increasing order results from a different angle, reducing the intensity of the diffracted X-rays as \(n\) increases.

In the given problem, the second-order diffraction is key to determining accurate crystal plane spacing. Since higher-order diffractions often require higher precision, correctly identifying and measuring them is crucial for obtaining reliable structural information.