Symmetry constraints shape crystal structures, dictating how atoms can arrange themselves. These rules, based on space group symmetry, determine which reflections show up in diffraction patterns and which are systematically absent. Understanding these constraints is key to figuring out crystal structures.

Structural motifs and symmetry-allowed distortions give crystals flexibility within their symmetry rules. But some changes are off-limits – they'd break the crystal's symmetry. Each crystal system has its own set of constraints, influencing how atoms can be arranged and coordinated.

Symmetry and Diffraction

Systematic Absences and Reflection Conditions

• Systematic absences occur when certain reflections are systematically missing from diffraction patterns due to symmetry elements in the crystal structure
• Reflection conditions describe the specific rules for which reflections will be present or absent based on the space group symmetry
• General reflection conditions apply to all reflections (hkl) in a given space group
• Special reflection conditions apply only to specific sets of reflections, such as (h00), (0k0), or (00l)
• Presence of screw axes leads to systematic absences along certain crystallographic directions
  • 21 screw axis along a results in absences for h00 reflections with h odd
  • 31 screw axis along c results in absences for 00l reflections with l not divisible by 3
• Glide planes cause systematic absences in certain zones of reflections
  • a-glide perpendicular to b results in absences for h0l reflections with h odd
  • n-glide perpendicular to c results in absences for hk0 reflections with h+k odd
• Centering of the unit cell produces systematic absences affecting all reflections
  • Body-centering (I) results in absences when h+k+l is odd
  • Face-centering (F) results in absences when h, k, and l are not all odd or all even

Space Group Determination

• Space group determination involves analyzing systematic absences and reflection conditions observed in diffraction data
• Process begins with identifying the Bravais lattice type based on unit cell parameters and centering-related absences
• Crystal system determined from the metric symmetry of the lattice
• Presence of screw axes and glide planes inferred from specific reflection conditions
• Laue class established by examining the overall symmetry of the diffraction pattern
• Combination of Bravais lattice, crystal system, and observed symmetry elements narrows down possible space groups
• Additional techniques may be required to distinguish between enantiomorphic pairs or to resolve ambiguities
  • Anomalous scattering experiments can differentiate between non-centrosymmetric and centrosymmetric space groups
  • Convergent beam electron diffraction (CBED) patterns provide information about point group symmetry
• Software tools (XPREP, SHELXT) assist in space group determination by analyzing systematic absences and intensity statistics

Structural Constraints

Structural Motifs and Symmetry-Allowed Distortions

• Structural motifs represent recurring arrangements of atoms or molecular units within crystal structures
• Common structural motifs include coordination polyhedra, organic functional groups, and inorganic frameworks
• Symmetry-allowed distortions maintain the overall symmetry of the crystal structure while allowing for local deviations
• Jahn-Teller distortions in octahedral complexes elongate or compress along one axis without breaking overall symmetry
  • Cu(II) complexes often exhibit elongation along the z-axis due to electronic configuration
• Perovskite structures (ABO3) allow for tilting of corner-sharing octahedra while maintaining overall symmetry
  • Glazer notation describes different tilt systems in perovskites (a+a+a+, a0b+b+, a-a-a-)
• Symmetry-allowed distortions can lead to phase transitions between different crystal structures
  • Ferroelectric materials undergo symmetry-lowering distortions that result in a polar structure (BaTiO3)

Symmetry-Forbidden Distortions and Crystal System Constraints

• Symmetry-forbidden distortions violate the symmetry requirements of the space group and are not allowed in the crystal structure
• Attempts to introduce symmetry-forbidden distortions result in a change of space group or symmetry breaking
• Crystal system constraints impose limitations on the possible arrangements of atoms within a given symmetry
• Cubic crystal system requires all three unit cell axes to be equal in length and all angles to be 90°
  • Restricts the possible coordination environments and packing arrangements
• Tetragonal crystal system allows for different c-axis length but maintains a=b and all angles 90°
  • Permits elongation or compression along one axis while maintaining fourfold symmetry
• Orthorhombic crystal system allows for three different axis lengths but all angles remain 90°
  • Provides more flexibility in atomic arrangements while maintaining orthogonality
• Lower symmetry crystal systems (monoclinic, triclinic) impose fewer constraints on atomic positions
  • Allow for more diverse structural arrangements and distortions
• Site symmetry of atomic positions must be consistent with the overall space group symmetry
  • Special positions have higher site symmetry and fewer degrees of freedom for atomic coordinates
  • General positions have lower site symmetry and more degrees of freedom for atomic coordinates
• Wyckoff positions describe the multiplicity and site symmetry of atomic positions within the unit cell
  • Higher symmetry space groups have more restricted sets of Wyckoff positions
  • Lower symmetry space groups offer more flexibility in atomic arrangements