Model Answer
0 min readIntroduction
X-ray diffraction (XRD) is a powerful non-destructive technique used to analyze the crystalline structure of materials. It relies on the constructive interference of X-rays scattered by atoms within a crystal lattice. The interaction between X-rays and the regularly spaced atoms in a crystal produces a diffraction pattern, which can be used to determine the arrangement of atoms, identify unknown materials, and quantify the amount of different phases in a mixture. Bragg’s Law, formulated in 1912 by William Henry Bragg and William Lawrence Bragg, provides the fundamental relationship governing this diffraction phenomenon, and is crucial for interpreting XRD data.
Derivation of Bragg’s Law
Consider a crystal with a set of parallel planes separated by a distance ‘d’. Let X-rays of wavelength ‘λ’ be incident on these planes at an angle ‘θ’. The X-rays are reflected from successive planes. For constructive interference to occur, the path difference between the rays reflected from adjacent planes must be an integer multiple of the wavelength.
Let’s consider two consecutive reflected rays. The path difference between these rays is given by:
Path Difference = 2d sin θ
For constructive interference, this path difference must be equal to nλ, where n is an integer (the order of reflection).
Therefore, the Bragg’s Law is expressed as:
nλ = 2d sin θ
Where:
- n = order of reflection (an integer, typically 1)
- λ = wavelength of the X-rays
- d = inter-planar spacing
- θ = angle of incidence (Bragg angle)
Calculation of Inter-planar Spacing
Given:
- θ = 50°
- λ = 1.5418 Å (for CuKα radiation)
- n = 1 (for the first-order reflection, which gives a sharp reflection)
Using Bragg’s Law (nλ = 2d sin θ), we can solve for ‘d’:
d = nλ / (2 sin θ)
d = (1 * 1.5418 Å) / (2 * sin 50°)
d = 1.5418 Å / (2 * 0.7660)
d = 1.5418 Å / 1.532
d ≈ 1.006 Å
Therefore, the inter-planar spacing in the crystal is approximately 1.006 Å.
Factors Affecting Diffraction Patterns
- Crystal Structure: Different crystal structures (e.g., FCC, BCC, HCP) produce unique diffraction patterns.
- Wavelength of X-rays: Shorter wavelengths provide higher resolution.
- Sample Preparation: Particle size and orientation affect the intensity and sharpness of the peaks.
- Instrumental Broadening: Imperfections in the instrument can broaden the diffraction peaks.
Applications of X-ray Diffraction
- Material Identification: Identifying unknown crystalline materials.
- Phase Analysis: Determining the composition of multi-phase materials.
- Crystallite Size Determination: Estimating the average size of crystallites in a material.
- Strain Analysis: Measuring the strain within a crystal lattice.
Conclusion
Bragg’s Law is a cornerstone of X-ray diffraction, providing a quantitative relationship between the wavelength of X-rays, the inter-planar spacing in a crystal, and the angle of incidence. The calculated inter-planar spacing of 1.006 Å for the given conditions provides valuable information about the crystal structure. XRD continues to be an indispensable tool in materials science, chemistry, and geology for characterizing crystalline materials and understanding their properties. Further advancements in XRD techniques, such as high-resolution XRD and synchrotron radiation XRD, are expanding its capabilities and applications.
Answer Length
This is a comprehensive model answer for learning purposes and may exceed the word limit. In the exam, always adhere to the prescribed word count.