Wave Diffraction and Bragg's Law Quiz

Wave Diffraction and the Reciprocal Lattice

• Diffraction of waves by crystals depends on crystal structure and incident radiation wavelength
• When wavelength is comparable to or smaller than lattice constant, diffracted beams appear in different directions
• Bragg's Law describes reflected waves from parallel atomic planes in a crystal
• A small fraction of radiation is reflected by each plane, behaving like a slightly silvered mirror
• X-ray energy remains unchanged during reflection (elastic scattering)

Bragg's Law

• Reflected waves from parallel atomic planes interfere constructively when path difference is an integral multiple of the wavelength.
• The formula is: nλ = 2dsinθ
  • n = integer order of reflection
  • λ = wavelength of radiation
  • d = distance between atomic planes
  • θ = angle of incidence

Diffraction and Crystal Structure

• X-rays scattered by atoms in a crystal interfere with each other, producing diffraction patterns
• Constructive interference occurs when waves are in phase, leading to intense diffracted beams
• Wavelength comparable to lattice spacing is essential for diffraction

Intensity of Diffraction

• In real crystals, not all reflection planes contribute equally.
• The intensity depends on the number of reflecting planes and factors like atom composition in the unit cell.
• The relative intensity of different diffraction orders is affected by these factors.

Fourier Analysis

• Crystals are periodic, meaning atom positions repeat regularly.
• Fourier analysis decomposes complex periodic functions into simpler sinusoidal components.
• The key idea in Fourier analysis is the use of sinusoidal functions to represent periodic phenomena

Periodicity and Reciprocal Lattice

• Local physical properties (e.g., charge density) in a crystal remain unchanged under translations.
• Fourier coefficients c_p and s_p in the x-direction form a periodic function.
• The reciprocal lattice vector is 2πp/a, and p is a positive integer.
• Points on the reciprocal lattice are directly related to the lattice spacing of the crystal.

Compact Form of Fourier Series

• Fourier series can be expressed using complex exponential functions with complex coefficients.
• For real functions, Fourier coefficients are related in a conjugate relationship.
• In three dimensions, we can expand a function using direct lattice vectors to obtain a similar expression for real functions.

Diffraction Conditions

• The reciprocal lattice vectors determine possible x-ray reflections
• Diffraction conditions are met when the reciprocal lattice vector is a multiple of the wavelength.
• G determines the possible x-ray reflections, where G is a reciprocal lattice vector. This dictates which reflections occur and the associated diffraction conditions.