Systematic absences and extinction rules are crucial for understanding crystal symmetry. These rules help identify specific symmetry elements in crystal structures by analyzing missing reflections in diffraction patterns.

Extinction rules determine which reflections are absent due to symmetry. By examining these absences, we can deduce the lattice type, screw axes, and glide planes present in a crystal, providing valuable insights into its space group.

Systematic Absences and Extinction Rules

Understanding Systematic Absences

• Systematic absences occur when specific reflections in a diffraction pattern disappear due to crystal symmetry
• Caused by destructive interference of X-ray waves scattered from symmetry-related atoms
• Provide crucial information about the crystal's space group and symmetry elements
• Differ from random absences caused by weak intensities or experimental limitations
• Can be observed in X-ray diffraction patterns, neutron diffraction, and electron diffraction

Extinction Rules and Their Applications

• Extinction rules determine which reflections are systematically absent for a given symmetry element
• Derived from the structure factor equation and symmetry operations
• Help identify specific symmetry elements present in the crystal structure
• Include rules for lattice centering, screw axes, and glide planes
• Vary depending on the type of symmetry element and its orientation in the unit cell

Analyzing Reflection Conditions and Diffraction Patterns

• Reflection conditions specify which reflections are allowed or forbidden based on Miller indices (hkl)
• Expressed using mathematical inequalities or congruences (h + k + l = 2n)
• Diffraction pattern analysis involves examining systematic absences to deduce symmetry elements
• Requires careful consideration of multiple zones and reflections to avoid misinterpretation
• Can be complicated by multiple contributing symmetry elements or pseudo-symmetry

Structure Factor and Lattice Centering

Structure Factor Fundamentals

• Structure factor (F) represents the amplitude and phase of diffracted X-rays from a crystal
• Calculated as the sum of scattering contributions from all atoms in the unit cell
• Expressed mathematically as $F_{hkl} = \sum_{j=1}^{N} f_j e^{2\pi i(hx_j + ky_j + lz_j)}$
• Depends on atomic positions (x, y, z), atomic scattering factors (f), and Miller indices (hkl)
• Determines the intensity of diffracted beams in X-ray crystallography

Lattice Centering and Its Effects

• Lattice centering introduces additional lattice points within the unit cell
• Types include primitive (P), body-centered (I), face-centered (F), and base-centered (A, B, C)
• Affects the structure factor by introducing phase shifts between scattered waves
• Results in systematic absences for specific combinations of Miller indices
• Extinction rules for centered lattices:
  • Body-centered (I): h + k + l = 2n
  • Face-centered (F): h, k, l all odd or all even
  • A-centered: k + l = 2n
  • B-centered: h + l = 2n
  • C-centered: h + k = 2n

Screw Axis and Glide Plane Absences

Screw Axis Symmetry and Absences

• Screw axes combine rotation with translation along the axis of rotation
• Notation n_m indicates an n-fold rotation with a translation of m/n of the repeat distance
• Generate systematic absences along specific directions in reciprocal space
• Extinction rules for common screw axes:
  • 2_1 axis parallel to b: 0k0 absent for k odd
  • 3_1 or 3_2 axis parallel to c: 00l absent for l not divisible by 3
  • 4_1, 4_2, or 4_3 axis parallel to c: 00l absent for l not divisible by 4
• Multiple screw axes can lead to more complex absence patterns

Glide Plane Symmetry and Related Absences

• Glide planes combine reflection with translation parallel to the reflection plane
• Types include a-glide, b-glide, c-glide, n-glide (diagonal), and d-glide (diamond)
• Produce systematic absences in specific planes of reciprocal space
• Extinction rules for common glide planes:
  • a-glide perpendicular to b: h0l absent for h odd
  • b-glide perpendicular to a: 0kl absent for k odd
  • c-glide perpendicular to a: h0l absent for l odd
  • n-glide perpendicular to c: hk0 absent for h + k odd
• Combination of multiple glide planes can result in more complex absence patterns
• Careful analysis required to distinguish between different types of glide planes