Question

Interplanar distances in crystal can be determined by equation (a) \(n \lambda=2 d \sin \theta\) (b) \(n=\lambda d \sin \theta\) (c) \(\frac{n}{\lambda}=2 d \sin \theta\)

Short answer

The correct formula for calculating interplanar distances in crystals is (a) \(n\lambda = 2d \sin(\theta)\)

Step-by-step solution

Analyze given options

Analyze each of the given options against the known formula for the interplanar distances in crystals, which is Bragg's law: \(n\lambda = 2d \sin(\theta)\)

Compare with Bragg's Law

Bragg's law is formulated as \(n\lambda = 2d \sin(\theta)\) where 'n' is the order of diffraction, \(\lambda\) is the wavelength, 'd' is the interplanar distance and \(\theta\) is the angle of incidence.

Identify the Correct Equation

By comparing the known formula with the given options, it is apparent that option (a) \(n\lambda = 2d \sin(\theta)\) is an exact match to Bragg's law.

Key concepts

Understanding Crystal Diffraction

Crystal diffraction is a fascinating phenomenon where a crystal structure interacts with an incoming X-ray beam. When X-rays strike a crystal, they are scattered by the atoms within the crystal lattice, creating a diffraction pattern. This pattern provides valuable information about the structure of the crystal.

The reason crystals cause X-ray diffraction is due to their regular and repetitive atomic arrangement, which allows them to act like a 3D diffraction grating. This orderly arrangement of atoms impacts the path of the X-ray beams, leading to constructive or destructive interference patterns.

• Constructive interference occurs when the path difference between waves is equal to a whole number multiple of their wavelength, resulting in intensified waves.
• Destructive interference happens when the path difference is a half number multiple of the wavelength, canceling out the waves.

These interactions occur at specific angles and provide insights into the crystal's atomic and molecular configuration.

Exploring Interplanar Spacing

Interplanar spacing, often denoted as 'd', is the distance between parallel planes of atoms in a crystal lattice. It is a critical dimension that helps to define the crystal's structure and is integral for determining how X-rays are diffracted by the crystal.

This spacing affects the angle at which X-rays are diffracted, indicating that precise measurements of 'd' are essential in crystallography. The position of the peaks in the diffraction pattern corresponds directly to these spacings within the crystal. These planes and their spacing are not arbitrary but are consistent with the periodic nature of the crystal lattice.

Understanding interplanar spacing helps scientists determine lattice parameters and the geometric arrangement of atoms. It's a foundational concept in both physics and materials science, providing insight into the detailed atomic structure of materials.

Understanding Bragg's Equation

Bragg's Equation, formulated by William Lawrence Bragg, is a fundamental law explaining how X-ray diffraction patterns are generated and analyzed. The equation is written as: \[ n\lambda = 2d \sin(\theta) \]
In this equation:

• 'n' represents the order of the diffraction, which is an integer indicating which peak in the diffraction pattern is being observed.
• '\lambda' is the wavelength of the incident X-ray beam.
• 'd' is the interplanar spacing of the crystal.
• '\theta' is the angle of incidence, or the angle at which the X-ray strikes the crystal plane.

Bragg's Law provides the condition for constructive interference in X-ray diffraction, meaning that the variable values of the equation will help determine if and how X-rays are scattered by the crystal lattice. By using this equation, scientists can decipher the crystal structure, gaining insights about the arrangement and spacing of atoms within the crystal. It serves as a bridge connecting the diffraction pattern to the physical structure of the crystal.