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X rays of wavelength \(0.12 \mathrm{~nm}\) are found to undergo secondorder reflection at a Bragg angle of \(28^{\circ}\) from a lithium fluoride crystal. What is the interplanar spacing of the reflecting planes in the crystal?

Short Answer

Expert verified
The interplanar spacing is approximately 0.255 nm.

Step by step solution

01

Understanding the Bragg's Law

Bragg's Law is used to calculate the interplanar spacing in a crystal lattice, and its formula is given by \( n \lambda = 2d \sin \theta \), where \( n \) is the order of reflection, \( \lambda \) is the wavelength of the X-rays, \( d \) is the interplanar spacing, and \( \theta \) is the Bragg angle.
02

Identifying Known Values

From the problem, we know the wavelength \( \lambda = 0.12 \) nm, the Bragg angle \( \theta = 28^{\circ} \), and the order of reflection \( n = 2 \). These will be used in Bragg’s Law.
03

Converting Angle to Radians

First, ensure your calculations account for the correct unit. Sometimes angles are preferred in radians for certain calculations, but here degree is consistent with typical use. If needed, convert \( \theta \) from degrees to radians using the formula \( \theta_{rad} = \theta_{degrees} \cdot \frac{\pi}{180} \). However, it is commonly calculated directly in degrees with sine value.
04

Simplifying Bragg's Law

Rearrange Bragg’s Law to solve for \( d \), the interplanar spacing: \( d = \frac{n \lambda}{2 \sin \theta} \). This formula will be used to calculate \( d \) using the given values.
05

Calculating the Interplanar Spacing

Substitute the known values into the simplified equation: \( d = \frac{2 \times 0.12 \text{ nm}}{2 \times \sin(28^{\circ})} = \frac{0.24 \text{ nm}}{2 \times \sin(28^{\circ})} \). Calculate \( \sin(28^{\circ}) \approx 0.4695 \).
06

Final Calculation

After approximating \( \sin(28^{\circ}) \), solve for \( d \): \( d = \frac{0.24 \text{ nm}}{2 \times 0.4695} \approx \frac{0.24}{0.939} \approx 0.255 \text{ nm} \). The interplanar spacing \( d \) is found to be approximately 0.255 nm.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

X-ray diffraction
X-ray diffraction is a fascinating phenomenon where X-rays are diffracted by the atomic planes within a crystal. This technique is crucial in studying the atomic structure of materials. When X-rays hit a crystal at a certain angle, they bounce off the layers of atoms, creating a pattern of interference. This pattern helps scientists understand the arrangement of the atoms within the crystal. This process is guided by Bragg's Law, which relates the diffraction angle, the wavelength of the X-rays, and the spacing between the atomic layers. X-ray diffraction is widely used in fields like crystallography, material science, and chemistry. It helps in determining the three-dimensional structures of compounds, which can be pivotal in drug design and other applications.
Interplanar spacing
Interplanar spacing refers to the distance between adjacent planes of atoms in a crystal lattice. This is an important parameter in the study of crystallography. By finding out the interplanar spacing, scientists can gain insight into the crystal's structure and its properties. Bragg's Law allows for the calculation of interplanar spacing, given the necessary parameters like the wavelength of X-rays and the angle of diffraction. This measurement is essential for identifying the type of crystal and understanding its physical characteristics. Accurate knowledge of interplanar spacing is crucial in applications involving semiconductors and metallurgy, among others.
Crystal lattice
A crystal lattice is a highly ordered and repeating arrangement of atoms, ions, or molecules in a crystal. This regular pattern extends in all three spatial dimensions and defines the structure of a crystal. Each point within a crystal lattice represents the position of a component such as an atom or molecule, which is repeated at consistent intervals. The geometry of a crystal lattice determines many physical properties of a material, including its strength, density, and conductivity. Understanding the arrangement of atoms within the lattice is essential for exploring how materials will behave under various conditions. Crystal lattices are fundamental in the development of new materials with specific attributes for industrial use.
Wavelength
Wavelength is a term used to describe the distance between successive peaks of a wave. In the context of X-ray diffraction, the wavelength is a critical factor that influences the angle at which X-rays are diffracted upon interacting with a crystal. It is usually measured in nanometers ( m) for X-rays, and typical wavelengths range between 0.01 to 10 m. Shorter wavelengths correspond to higher energy X-rays and are particularly useful for probing the atomic structure of materials. The precise measurement of wavelength is crucial in X-ray diffraction experiments, as it directly impacts the calculation of the interplanar spacing in a crystal lattice. Understanding the concept of wavelength allows scientists to tailor their experiments to various types of crystals and materials.

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Most popular questions from this chapter

A double-slit system with individual slit widths of \(0.030 \mathrm{~mm}\) and a slit separation of \(0.18 \mathrm{~mm}\) is illuminated with \(500 \mathrm{~nm}\) light directed perpendicular to the plane of the slits. What is the total number of complete bright fringes appearing between the two first-order minima of the diffraction pattern? (Do not count the fringes that coincide with the minima of the diffraction pattern.)

Suppose that the central diffraction envelope of a double-slit diffraction pattern contains 11 bright fringes and the first diffraction minima eliminate (are coincident with) bright fringes. How many bright fringes lie between the first and second minima of the diffraction envelope?

Light of wavelength \(633 \mathrm{~nm}\) is incident on a narrow slit. The angle between the first diffraction minimum on one side of the central maximum and the first minimum on the other side is \(1.20^{\circ} .\) What is the width of the slit?

Nuclear-pumped x-ray lasers are seen as a possible weapon to destroy ICBM booster rockets at ranges up to \(2000 \mathrm{~km}\). One limitation on such a device is the spreading of the beam due to diffraction, with resulting dilution of beam intensity. Consider such a laser operating at a wavelength of \(1.40 \mathrm{~nm}\). The element that emits light is the end of a wire with diameter \(0.200 \mathrm{~mm}\). (a) Calculate the diameter of the central beam at a target \(2000 \mathrm{~km}\) away from the beam source. (b) What is the ratio of the beam intensity at the target to that at the end of the wire? (The laser is fired from space, so neglect any atmospheric absorption.)

The wings of tiger beetles (Fig. \(36-41\) ) are colored by interference due to thin cuticle-like layers. In addition, these layers are arranged in patches that are \(60 \mu \mathrm{m}\) across and produce different colors. The color you see is a pointillistic mixture of thin-film interference colors that varies with perspective. Approximately what viewing distance from a wing puts you at the limit of resolving the different colored patches according to Rayleigh's criterion? Use \(550 \mathrm{~nm}\) as the wavelength of light and \(3.00 \mathrm{~mm}\) as the diameter of your pupil.

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