Introduction to Crystallography: Lecture 5 — Plane and Space Groups

Scientific concepts / discoveries / nature phenomena in the subtitles

Crystallography conventions

• Plane groups (2D) vs. space groups (3D) How plane groups and space groups are represented and interpreted in crystallography.

• Representing and reading space-group symbols

  • Standard symbols follow a convention where a capital centering/setting letter is paired with symmetry entries.
  • The symmetry entries encode the crystal system and the relevant symmetry elements.

Crystal systems (7) and their symmetry constraints

• Triclinic

  • Generally only an inversion center.
  • No rotations higher than 1: only “1” or “(\bar{1})” indicated.

• Monoclinic

  • Has one unique axis (conventionally (\beta \neq 90^\circ)).
  • Rotation symmetry can exist, but only in one direction (per the convention).

• Orthorhombic

  • All angles are 90°.
  • Allows rotational symmetry along all three directions.
  • Cannot omit entries corresponding to symmetry in the three directions.

• Rhombic (rhombohedral/trigonal context in the subtitles)

  • Characterized by threefold symmetry.
  • Recognized by the presence of 3 or (\bar{3}).

• Tetragonal

  • Recognized by 4/ (\bar{4})-type symmetry.
  • The fourfold axis lies along the unique axis (conventionally the “C” direction).

• Hexagonal

  • Contains 6/ (\bar{6})-type symmetry.
  • Noted that centering symbols do not appear for hexagonal in this convention.

• Cubic

  • Recognized via 3 or (\bar{3}) appearing in the correct entry position.
  • Rotation logic involves body diagonals.

Space-group structure encoded by symbols

• Hermann–Mauguin / International space-group symbols

  • Also described as “standard crystal graphic symbols.”

• Implied symmetry operations

  • Short symbols may encode additional operations through conventions and generators.

• Slash notation (e.g., “/m”)

  • Indicates a rotation axis paired with a plane perpendicular to it.
  • Both appear together in the same symmetry position.

• Omission conventions

  • Some entries may be left out if they’re implied or redundant.

How to identify symmetry directions from symbol entries

• Axis/direction meaning

  • Entries correspond to directions tied to the unit-cell geometry and symmetry operations.
  • Includes distinctions between:
    • unit-cell axes
    • body diagonals
    • diagonals in planes

• Rhombohedral vs. trigonal/hexagonal settings

  • Rhombohedral structures can be represented in an alternate “hexagonal/trigonal” setting.
  • The axes/angles differ, but they represent the same underlying structure.

Space-group diagrams

• What diagrams show

  • How symmetry elements’ locations and orientations are represented.

• Projection conventions

  • Diagrams are typically projected into the drawing plane.
  • Elements not in the plane are marked using a projection indicator (subtitles mention “P”) and/or height.

• Fractional translations/distances

  • Diagram distances and symmetry translations are often given as fractions of the unit cell length (e.g., quarter-cell shifts), not absolute distances.
  • Reason: fractional coordinates are relative, comparable even if unit-cell size changes.

Asymmetric unit and Wyckoff positions

• Asymmetric unit

  • The smallest region that generates the full space group via symmetry operations.
  • Can be smaller than the unit cell.
  • Can have non-rectangular/irregular shapes in higher symmetry cases.

• Wyckoff notation / Wyckoff letters

  • Different sites have different symmetry constraints.
  • Multiplicity: how many equivalent positions are generated in the unit cell for an atom on that Wyckoff site.

• General position

  • A site not lying on any symmetry element.
  • Coordinates are arbitrary (often written as (x,y,z)).

Symmetry operations and chirality indicators in diagrams

• Chiral objects

  • Diagrams may include handedness indicators (e.g., commas/teardrops) to show one chirality.

• “+ / −” markers

  • “+” and “−” indicate relative placement above vs. below the projection plane.
  • Handedness can change under symmetry operations.

• Complex diagram interpretation

  • When two objects appear behind each other, they may show opposite handedness and/or different relative height.

Origins in space groups

• Choosing the origin

  • If the space group is centrosymmetric: origin is placed at an inversion center.
  • If non-centrosymmetric: origin is placed at a point of highest symmetry:
    • highest rotation axis,
    • intersections of symmetry elements,
    • or special cases involving screw axes.

• Why origin choice matters computationally

  • Choosing the wrong origin can scramble atomic positions and prevent refinement from working.

Generators and symmetry tables

• What space-group tables list
  • Symmetry operations (often as matrices/actions).
  • Generator sets: minimal operations needed to regenerate the full group.
  • Sites/atomic positions with coordinates, plus reflection conditions (mentioned without deep detail in the subtitles).

Subgroups, supergroups, and symmetry reduction

• Subgroup relations

  • A subgroup contains a subset of the parent group’s symmetry elements.
  • Often organized as chains of decreasing symmetry.
  • “Maximum subgroups” are referenced.

• Ways symmetry can change

  • Removing point symmetry elements (e.g., dropping rotation axes or inversion centers) while keeping translations equivalent.
  • Changing translation symmetry, e.g., via unit-cell changes that alter screw-axis relationships.
  • Classification concepts mentioned:
    • k-subgroups (translation-related changes)
    • t-subgroups / “t-equivalent” (wording used in the subtitles)
    • Subtitles also mention an opposite relation described as a “silver group.”

• Connection to phase transitions

  • Symmetry can lower through small distortions as temperature changes.

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Methodology / workflow outlined

• How to read a space-group symbol

  1. Identify the capital centering/setting letter (e.g., P, R, I, F; subtitles also mention non-standard variants).
  2. Use the following symmetry entries (typically 1–3 symbols) to deduce:
     • the crystal system
     • the types/axes of rotational symmetry and constraints
  3. Apply recognition rules:
     • Triclinic: only “1” or “(\bar{1})” after the centering/setting.
     • Rhombohedral: presence of 3 or (\bar{3}).
     • Tetragonal: presence of 4/ (\bar{4}).
     • Hexagonal: presence of 6 (and the convention that centering symbols don’t appear there).
     • Cubic: the critical role of 3 / (\bar{3}) placement in the symbol.

• How to interpret symmetry direction meaning in the symbol

  • The first entry is tied to a unit-cell-related direction (system-specific convention).
  • The second entry is tied to a different crystallographic direction (e.g., a body diagonal in cubic).
  • The third entry (when present) corresponds to a direction diagonal to the unit-cell axes.

• How to use space-group diagrams

  • Recognize diagrams as projected drawings.
  • For elements not in the plane:
    • use projection indicators and/or height fractions (e.g., (1/4) of unit cell length).
  • Visualize symmetry by applying the shown symmetry elements (e.g., mirror/rotation/glide).

• How to determine atomic positions from tables

  1. Start from an asymmetric unit.
  2. Apply the listed symmetry operations (generators and combinations).
  3. Use Wyckoff letters to assign symmetry-constrained sites and their multiplicity.
  4. For atoms not on symmetry elements, use the general position definition.

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Researchers / sources featured

• Hermann–Mauguin

  • Referenced as the standard space-group symbol system (with a pronunciation note only).

• “International Tables”

  • Mentioned as a source for tabulated symmetry conventions and representations.

• “The Big Blue Book”

  • Referred to by the instructor as a primary reference for space-group/plane-group symbol meaning (no individual author named in the subtitles).