Question

Name of equation related to X-ray diffraction by crystals is: (a) Laue equation (b) Bragg's equation (c) Lewis equation

Short answer

The equation associated with x-ray diffraction by crystals is Bragg's equation.

Step-by-step solution

Analyze the Options

Take a brief look at all the options available. Laue equation is utilized for the diffraction from a three-dimensional periodic object. The Lewis structure is a simplified way of representing the bondings in molecules or ions. Bragg's equation is used to determine the spacings between planes in a crystal from the angles at which an incident x-ray beam is diffracted.

Identify the Correct Equation

From the understanding gained in step 1, it's clear that Bragg's equation is the one associated with x-ray diffraction by crystals. Lewis structure is totally irrelevant here as it is related to chemistry and molecular structure, and Laue equation, despite being related to diffraction, is associated with a three-dimensional object and not specifically with crystals.

Formulate the Answer

After analyzing the provided options and their relevance to the phenomenon in question, the answer to the equation related to x-ray diffraction by crystals is Bragg's equation.

Key concepts

Bragg's equation

Understanding the principles behind X-ray diffraction by crystals is crucial for the field of crystallography. One of the key concepts is Bragg's equation, which is fundamentally important in interpreting X-ray diffraction patterns. This equation relates the wavelength of incident X-rays and the crystal structure.

The equation is expressed as: \[ n\textbackslash\textlambda = 2d \textbackslash sin(\theta) \] where:

• \( n \) is the order of the diffraction pattern,
• \( \lambda \) is the wavelength of the X-rays,
• \( d \) is the distance between the crystal planes, and
• \( \theta \) is the angle of incidence at which the diffraction peak occurs.

In simple terms, Bragg's equation helps us calculate the position and intensity of 'spots' in a diffraction pattern which reveal the positions of the planes within a crystal. When X-rays hit a crystal structure, they are scattered in various directions. If these scattered waves interfere constructively, they create a diffracted ray that can be detected and analyzed.

By adjusting the angle \( \theta \), we can find specific points where the path difference between rays scattered from successive crystal planes equals an integer multiple of the X-ray wavelength, satisfying the condition for constructive interference.

Laue equation

While Bragg's equation is widely utilized for crystal analysis, the Laue equation also plays a significant role in understanding diffraction patterns, particularly when dealing with more complex crystal orientations and structures. The Laue equation is used to describe diffraction in three-dimensional patterns.

The Laue conditions can be given as a set of equations:\[ \textbackslash vec{a}^*_i \textbackslash cdot \textbackslash vec{r} = h,\textbackslash vec{b}^*_i \textbackslash cdot \textbackslash vec{r} = k, \textbackslash vec{c}^*_i \textbackslash cdot \textbackslash vec{r} = l \] where \( \textbackslash vec{a}^*_i, \textbackslash vec{b}^*_i, \) and \( \textbackslash vec{c}^*_i \) are the reciprocal lattice vectors and \( h, k, l \) are the Laue indices. The vector \( \textbackslash vec{r} \) represents the direction of the incoming X-rays.

The Laue equations are essential when the X-ray beam is polychromatic or when the crystal is oriented in a general, rather than a specific, direction. These equations allow us to map the three-dimensional structure of the crystal, which is key in fields such as mineralogy and materials science.

Crystallography

Crystallography is the scientific study of crystals and their atomic structure. It largely revolves around understanding how the atoms in a crystal are arranged and how they interact with various forms of electromagnetic radiation, like X-rays. Techniques like X-ray diffraction form the backbone of crystallography, as they allow scientists to probe the internal structure of crystals without physically breaking them apart.

To identify the molecular structure, crystallographers grow crystals of a material and then illuminate them with X-rays. By analyzing the pattern of X-ray diffraction, scientists can infer the positions of the atoms within the crystal. This process has been pivotal in numerous scientific achievements including the uncovering of the double helix structure of DNA.

Crystallography has applications across many fields, including chemistry, physics, biology, and materials science. It provides critical insights into the properties of materials, such as their stability, reactivity, and how they can be improved for various applications. The technique continues to evolve with advancements in technology, computational methods, and the development of new diffraction methods like neutron and electron diffraction.