The Ewald sphere and reciprocal lattice are key concepts in understanding diffraction patterns in crystals. They provide a visual way to represent the conditions needed for X-rays to scatter off crystal planes, helping us predict where diffraction spots will appear.

These tools connect the real crystal structure to the patterns we see in diffraction experiments. By using the Ewald sphere and reciprocal lattice, we can figure out important info about a crystal's structure from its diffraction pattern.

Ewald Sphere and Reciprocal Lattice

Fundamental Concepts of Reciprocal Space

• Reciprocal space represents the Fourier transform of real space in crystallography
• Reciprocal lattice consists of points corresponding to planes in the real crystal lattice
• Reciprocal lattice vector $\mathbf{G}$ connects the origin to a point in the reciprocal lattice
• Relationship between real space lattice vectors $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ and reciprocal lattice vectors \mathbf{a^*}$, $\mathbf{b^*}, \mathbf{c^*}$ defined by equations such as $\mathbf{a^*} = \frac{\mathbf{b} \times \mathbf{c}}{\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})}
• Reciprocal lattice vector magnitude inversely proportional to the spacing between planes in real space

Ewald Sphere Construction and Properties

• Ewald sphere serves as a geometric representation of diffraction conditions
• Sphere constructed with radius $1/\lambda$, where $\lambda$ represents the wavelength of incident radiation
• Center of the Ewald sphere positioned at the tip of the incident wavevector $\mathbf{k}_0$
• Origin of reciprocal space placed at the point where $\mathbf{k}_0$ intersects the sphere
• Diffraction occurs when a reciprocal lattice point lies on the surface of the Ewald sphere
• Ewald sphere radius changes with different wavelengths, affecting observable diffraction patterns

Diffraction Conditions

Laue Condition and Scattering Vector

• Laue condition defines the requirement for constructive interference in diffraction
• Mathematically expressed as $\mathbf{k} - \mathbf{k}_0 = \mathbf{G}$, where $\mathbf{k}$ represents the scattered wavevector
• Scattering vector $\mathbf{Q}$ defined as the difference between scattered and incident wavevectors: $\mathbf{Q} = \mathbf{k} - \mathbf{k}_0$
• Diffraction occurs when the scattering vector equals a reciprocal lattice vector
• Magnitude of scattering vector related to scattering angle $\theta$ by $|\mathbf{Q}| = \frac{4\pi}{\lambda} \sin(\theta)$

Brillouin Zones and Diffraction Analysis

• Brillouin zones represent regions in reciprocal space bounded by planes that bisect reciprocal lattice vectors
• First Brillouin zone contains all points in reciprocal space closer to the origin than to any other reciprocal lattice point
• Higher-order Brillouin zones defined by increasing distance from the origin
• Brillouin zone boundaries correspond to positions where strong diffraction occurs
• Analysis of Brillouin zones aids in understanding electronic band structures and phonon dispersion in crystals
• Relationship between Brillouin zones and allowed electron states in solids (energy bands)