Problem 88
Question
The longest wavelength that can be analysed by a sodium chloride crystal of spacing \(d=2.82 \AA\) in the second order is (a) \(2.82 \AA\) (b) \(5.64 \overline{\mathrm{A}}\) (c) \(8.46 \AA\) (d) \(11.28 \AA\)
Step-by-Step Solution
Verified Answer
The longest wavelength is 5.64 \( \AA \). (Option b)
1Step 1: Understand the Problem
We need to determine the longest possible wavelength that can be analyzed by a sodium chloride crystal with a given spacing when using the second-order reflection.
2Step 2: Apply Bragg's Law
Bragg's Law is given by the equation: \[ n\lambda = 2d\sin\theta \] where \( n \) is the order of reflection, \( \lambda \) is the wavelength, \( d \) is the spacing between crystal planes, and \( \theta \) is the angle of incidence. We need to find \( \lambda \) when \( n = 2 \).
3Step 3: Identify the Maximum Sin Value
The maximum value of \( \sin \theta \) is 1, which corresponds to the maximum wavelength. Substitute this into Bragg's Law to find the maximum wavelength.
4Step 4: Calculate Maximum Wavelength
Substitute \( n = 2 \), \( d = 2.82 \AA\), and \( \sin \theta = 1 \): \[ \lambda = \frac{2 \times 2.82 \AA}{2} = 2.82 \times 2 \AA = 5.64 \AA \]
5Step 5: Find the Correct Answer
The calculated maximum wavelength is \( 5.64 \AA \), which corresponds to option (b).
Key Concepts
Crystal SpacingWavelength CalculationOrder of Reflection
Crystal Spacing
Crystal spacing, often denoted as \(d\), refers to the distance between consecutive planes of atoms in a crystal lattice. It is one of the fundamental properties in crystallography. Crystals are known for their highly ordered structure, which means the atoms are arranged in a repeating pattern over long distances. This repeating pattern forms planes that reflect incident radiation. Understanding the spacing between these planes is crucial for analyzing how crystals interact with electromagnetic waves, such as X-rays and neutrons.
The spacing can be thought of as the 'resolution' at which a crystal can diffract or scatter these waves. If the spacing is small, it implies that the crystal can scatter shorter wavelengths more effectively. Conversely, larger spacing can handle longer wavelengths. This is central to measuring and analyzing the wavelengths of electromagnetic radiation passing through or reflecting off a crystal.
Ultimately, crystal spacing plays a vital role in numerous applications, such as determining the structure of materials through techniques like X-ray crystallography. By employing Bragg's Law, scientists can calculate the spacing between crystal planes using measured diffraction angles and analyze the crystal structure accordingly.
The spacing can be thought of as the 'resolution' at which a crystal can diffract or scatter these waves. If the spacing is small, it implies that the crystal can scatter shorter wavelengths more effectively. Conversely, larger spacing can handle longer wavelengths. This is central to measuring and analyzing the wavelengths of electromagnetic radiation passing through or reflecting off a crystal.
Ultimately, crystal spacing plays a vital role in numerous applications, such as determining the structure of materials through techniques like X-ray crystallography. By employing Bragg's Law, scientists can calculate the spacing between crystal planes using measured diffraction angles and analyze the crystal structure accordingly.
Wavelength Calculation
For wavelength calculation in the context of crystal analysis, Bragg's Law is the fundamental equation. This law states: \[ n\lambda = 2d\sin\theta \] where:
\[ \lambda = \frac{2d\sin\theta}{n} \]
In the original problem, you analyze the longest wavelength possible by choosing the maximum value for sin(θ), which is 1, because sin(θ) ranges from -1 to 1. This simplifies the equation significantly, allowing for straightforward calculation of wavelength.
Thus, by substituting \(\sin\theta = 1\), \(n = 2\), and \(d = 2.82 \text{\AA}\), you calculate:
\[ \lambda = \frac{2 \times 2.82}{2} = 5.64 \text{\AA}\]
This provides a clear understanding of how wavelength can be calculated using measurements of crystal properties and Bragg's Law.
- \(n\) is the order of reflection, which signifies the number of wavelengths that fit into the path difference between reflected waves.
- \(\lambda\) represents the wavelength of the incident wave.
- \(d\) is the crystal spacing.
- \(\theta\) is the angle of incidence which is also the angle of reflection.
\[ \lambda = \frac{2d\sin\theta}{n} \]
In the original problem, you analyze the longest wavelength possible by choosing the maximum value for sin(θ), which is 1, because sin(θ) ranges from -1 to 1. This simplifies the equation significantly, allowing for straightforward calculation of wavelength.
Thus, by substituting \(\sin\theta = 1\), \(n = 2\), and \(d = 2.82 \text{\AA}\), you calculate:
\[ \lambda = \frac{2 \times 2.82}{2} = 5.64 \text{\AA}\]
This provides a clear understanding of how wavelength can be calculated using measurements of crystal properties and Bragg's Law.
Order of Reflection
In crystallography, the order of reflection, denoted by \(n\), is an integer value that indicates the number of wavelengths that fit into the path difference caused by reflection from consecutive lattice planes. It essentially counts how many complete wavelengths are accommodated in the reflection process.
Bragg's Law incorporates the order of reflection as an essential variable. A higher order of reflection means that the incident wave is diffracting off multiple planes within the crystal. This increases the potential path difference, resulting in the constructive interference needed to observe a diffraction pattern.
For example, in the original exercise, the provided \(n = 2\) specifies a second-order reflection. This means the analysis looks at the scenario where two complete wavelengths fit into the path difference along with the increased path length due to extra lattice planes. Higher orders can provide different insights into the structural characteristics of materials, depending on how the resulting diffracted beams are analyzed.
The concept of order of reflection underscores how versatile crystallography is in probing the internal structure of materials. By analyzing different orders, scientists can extract more detailed information about the periodicity and other properties of crystals.
Bragg's Law incorporates the order of reflection as an essential variable. A higher order of reflection means that the incident wave is diffracting off multiple planes within the crystal. This increases the potential path difference, resulting in the constructive interference needed to observe a diffraction pattern.
For example, in the original exercise, the provided \(n = 2\) specifies a second-order reflection. This means the analysis looks at the scenario where two complete wavelengths fit into the path difference along with the increased path length due to extra lattice planes. Higher orders can provide different insights into the structural characteristics of materials, depending on how the resulting diffracted beams are analyzed.
The concept of order of reflection underscores how versatile crystallography is in probing the internal structure of materials. By analyzing different orders, scientists can extract more detailed information about the periodicity and other properties of crystals.
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