(International Tables) Space-group symmetry

International Tables for Crystallography
Volume A: Space-group symmetry

Second online edition (2016) ISBN: 978-0-470-97423-0 doi: 10.1107/97809553602060000114

Edited by M. I. Aroyo

Foreword to the Sixth Edition  (p. xv) | html | pdf |
Carolyn Pratt Brock
Preface  (pp. xvii-xix) | html | pdf |
Mois I. Aroyo
Symbols for crystallographic items used in this volume  (pp. xx-xxi) | html | pdf |
M. I. Aroyo

Introduction to space-group symmetry
- 1.1. A general introduction to groups (pp. 2-11) | html | pdf | chapter contents |
  B. Souvignier
  - 1.1.1. Introduction (p. 2) | html | pdf |
  - 1.1.2. Basic properties of groups (pp. 2-4) | html | pdf |
  - 1.1.3. Subgroups (pp. 4-5) | html | pdf |
  - 1.1.4. Cosets (pp. 5-6) | html | pdf |
  - 1.1.5. Normal subgroups, factor groups (pp. 6-7) | html | pdf |
  - 1.1.6. Homomorphisms, isomorphisms (pp. 7-9) | html | pdf |
  - 1.1.7. Group actions (pp. 9-10) | html | pdf |
  - 1.1.8. Conjugation, normalizers (pp. 10-11) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 1.1.2.1. Symmetry group 2mm of a rectangle (p. 3) | html | pdf |
    - Fig. 1.1.2.2. Symmetry group 3m of an equilateral triangle (p. 3) | html | pdf |
    - Fig. 1.1.2.3. Symmetry group 4mm of the square (p. 3) | html | pdf |
    - Fig. 1.1.3.1. Subgroup diagram for the symmetry group 3m of an equilateral triangle (p. 4) | html | pdf |
    - Fig. 1.1.3.2. Subgroup diagram for the symmetry group 2mm of a rectangle (p. 5) | html | pdf |
    - Fig. 1.1.3.3. Subgroup diagram for the symmetry group 4mm of the square (p. 5) | html | pdf |
    - Fig. 1.1.5.1. Symmetry group of an eightfold star (p. 7) | html | pdf |
    - Fig. 1.1.6.1. Schematic description of a homomorphism (p. 8) | html | pdf |
    - Fig. 1.1.6.2. Cyclic group of order 6 embedded in the group of the unit circle (p. 8) | html | pdf |
    - Fig. 1.1.7.1. Stabilizers in the symmetry group 4mm of the square (p. 10) | html | pdf |
- 1.2. Crystallographic symmetry (pp. 12-21) | html | pdf | chapter contents |
  H. Wondratschek and M. I. Aroyo
  - 1.2.1. Crystallographic symmetry operations (pp. 12-13) | html | pdf |
  - 1.2.2. Matrix description of symmetry operations (pp. 13-19) | html | pdf |
    - 1.2.2.1. Matrix–column presentation of isometries (pp. 13-15) | html | pdf |
      - 1.2.2.1.1. Shorthand notation of matrix–column pairs (pp. 13-15) | html | pdf |
    - 1.2.2.2. Combination of mappings and inverse mappings (p. 15) | html | pdf |
    - 1.2.2.3. Matrix–column pairs and (3 + 1) × (3 + 1) matrices (p. 16) | html | pdf |
    - 1.2.2.4. The geometric meaning of (W, w) (pp. 16-18) | html | pdf |
    - 1.2.2.5. Determination of matrix–column pairs of symmetry operations (pp. 18-19) | html | pdf |
  - 1.2.3. Symmetry elements (pp. 19-21) | html | pdf |
  - References | html | pdf |
  - Tables
    - Table 1.2.2.1. Matrices for point-group symmetry operations and orientation of corresponding geometric elements, referred to a cubic, tetragonal, orthorhombic, monoclinic, triclinic or rhombohedral coordinate system (p. 14) | html | pdf |
    - Table 1.2.2.2. Matrices for point-group symmetry operations and orientation of corresponding geometric elements, referred to a hexagonal coordinate system (p. 15) | html | pdf |
    - Table 1.2.3.1. Symmetry elements in point and space groups (p. 20) | html | pdf |
- 1.3. A general introduction to space groups (pp. 22-41) | html | pdf | chapter contents |
  B. Souvignier
  - 1.3.1. Introduction (p. 22) | html | pdf |
  - 1.3.2. Lattices (pp. 22-28) | html | pdf |
    - 1.3.2.1. Basic properties of lattices (pp. 22-23) | html | pdf |
    - 1.3.2.2. Metric properties (pp. 23-24) | html | pdf |
    - 1.3.2.3. Unit cells (p. 24) | html | pdf |
    - 1.3.2.4. Primitive and centred lattices (pp. 24-27) | html | pdf |
    - 1.3.2.5. Reciprocal lattice (pp. 27-28) | html | pdf |
  - 1.3.3. The structure of space groups (pp. 28-31) | html | pdf |
    - 1.3.3.1. Point groups of space groups (pp. 28-29) | html | pdf |
    - 1.3.3.2. Coset decomposition with respect to the translation subgroup (pp. 29-31) | html | pdf |
    - 1.3.3.3. Symmorphic and non-symmorphic space groups (p. 31) | html | pdf |
  - 1.3.4. Classification of space groups (pp. 31-41) | html | pdf |
    - 1.3.4.1. Space-group types (pp. 31-33) | html | pdf |
    - 1.3.4.2. Geometric crystal classes (pp. 33-34) | html | pdf |
    - 1.3.4.3. Bravais types of lattices and Bravais classes (pp. 34-37) | html | pdf |
    - 1.3.4.4. Other classifications of space groups (pp. 37-41) | html | pdf |
      - 1.3.4.4.1. Arithmetic crystal classes (pp. 37-39) | html | pdf |
      - 1.3.4.4.2. Lattice systems (p. 39) | html | pdf |
      - 1.3.4.4.3. Crystal systems (pp. 39-40) | html | pdf |
      - 1.3.4.4.4. Crystal families (pp. 40-41) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 1.3.2.1. Conventional basis $[{\bf a}, {\bf b}]$ and a non-conventional basis $[{\bf a}', {\bf b}']$ for the square lattice (p. 23) | html | pdf |
    - Fig. 1.3.2.2. Voronoï domains and primitive unit cells for a rectangular lattice (a) and an oblique lattice (b) (p. 24) | html | pdf |
    - Fig. 1.3.2.3. Primitive rectangular lattice (only the filled nodes) and centred rectangular lattice (filled and open nodes) (p. 25) | html | pdf |
    - Fig. 1.3.2.4. Primitive cell (dashed line) and centred cell (solid lines) for the centred rectangular lattice (p. 25) | html | pdf |
    - Fig. 1.3.4.1. Classification levels for three-dimensional space groups (p. 32) | html | pdf |
    - Fig. 1.3.4.2. Space-group diagram of (left) and its reflection in the plane z = 0 (right) (p. 33) | html | pdf |
  - Tables
    - Table 1.3.3.1. Automorphism groups of two-dimensional primitive lattices (p. 29) | html | pdf |
    - Table 1.3.3.2. Automorphism groups of three-dimensional primitive lattices (p. 30) | html | pdf |
    - Table 1.3.3.3. Right-coset decomposition of $[{\cal G}]$ relative to $[{\cal T}]$ (p. 30) | html | pdf |
    - Table 1.3.4.1. Lattice systems in three-dimensional space (p. 39) | html | pdf |
    - Table 1.3.4.2. Crystal systems in three-dimensional space (p. 39) | html | pdf |
    - Table 1.3.4.3. Distribution of space-group types in the hexagonal crystal family (p. 40) | html | pdf |
- 1.4. Space groups and their descriptions (pp. 42-73) | html | pdf | chapter contents |
  B. Souvignier, H. Wondratschek, M. I. Aroyo, G. Chapuis and A. M. Glazer
  - 1.4.1. Symbols of space groups (pp. 42-50) | html | pdf |
    H. Wondratschek
    - 1.4.1.1. Introduction (p. 42) | html | pdf |
    - 1.4.1.2. Space-group numbers (p. 42) | html | pdf |
    - 1.4.1.3. Schoenflies symbols (pp. 42-43) | html | pdf |
      - 1.4.1.3.1. Schoenflies symbols of the crystal classes (pp. 42-43) | html | pdf |
      - 1.4.1.3.2. Schoenflies symbols of the space groups (p. 43) | html | pdf |
    - 1.4.1.4. Hermann–Mauguin symbols of the space groups (pp. 43-48) | html | pdf |
      - 1.4.1.4.1. Introduction (pp. 43-44) | html | pdf |
      - 1.4.1.4.2. General aspects (pp. 44-45) | html | pdf |
      - 1.4.1.4.3. Triclinic space groups (pp. 45-46) | html | pdf |
      - 1.4.1.4.4. Monoclinic space groups (p. 46) | html | pdf |
      - 1.4.1.4.5. Orthorhombic space groups (pp. 46-47) | html | pdf |
      - 1.4.1.4.6. Tetragonal space groups  (p. 47) | html | pdf |
        1.4.1.4.6.1. Tetragonal space groups with one symmetry direction  (p. 47) | html | pdf |
        1.4.1.4.6.2. Tetragonal space groups with three symmetry directions  (p. 47) | html | pdf |
      - 1.4.1.4.7. Trigonal, hexagonal and rhombohedral space groups  (pp. 47-48) | html | pdf |
        1.4.1.4.7.1. Trigonal space groups  (pp. 47-48) | html | pdf |
        1.4.1.4.7.2. Hexagonal space groups  (p. 48) | html | pdf |
        1.4.1.4.7.3. Rhombohedral space groups  (p. 48) | html | pdf |
      - 1.4.1.4.8. Cubic space groups (p. 48) | html | pdf |
    - 1.4.1.5. Hermann–Mauguin symbols of the plane groups (pp. 48-49) | html | pdf |
    - 1.4.1.6. Sequence of space-group types (pp. 49-50) | html | pdf |
  - 1.4.2. Descriptions of space-group symmetry operations (pp. 50-59) | html | pdf |
    M. I. Aroyo, G. Chapuis, B. Souvignier and A. M. Glazer
    - 1.4.2.1. Symbols for symmetry operations (pp. 50-51) | html | pdf |
    - 1.4.2.2. Seitz symbols of symmetry operations (pp. 51-53) | html | pdf |
    - 1.4.2.3. Symmetry operations and the general position (pp. 53-55) | html | pdf |
    - 1.4.2.4. Additional symmetry operations and symmetry elements (pp. 55-56) | html | pdf |
    - 1.4.2.5. Space-group diagrams (pp. 56-59) | html | pdf |
  - 1.4.3. Generation of space groups (pp. 59-61) | html | pdf |
    H. Wondratschek
    - 1.4.3.1. Selected order for non-translational generators (pp. 60-61) | html | pdf |
  - 1.4.4. General and special Wyckoff positions (pp. 61-67) | html | pdf |
    B. Souvignier
    - 1.4.4.1. Crystallographic orbits (pp. 61-62) | html | pdf |
    - 1.4.4.2. Wyckoff positions (pp. 62-64) | html | pdf |
    - 1.4.4.3. Wyckoff sets (pp. 64-66) | html | pdf |
    - 1.4.4.4. Eigensymmetry groups and non-characteristic orbits (pp. 66-67) | html | pdf |
  - 1.4.5. Sections and projections of space groups (pp. 67-73) | html | pdf |
    B. Souvignier
    - 1.4.5.1. Introduction (pp. 67-68) | html | pdf |
    - 1.4.5.2. Sections (pp. 68-71) | html | pdf |
    - 1.4.5.3. Projections (pp. 71-73) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 1.4.1.1. Symmetry-element diagrams of some point groups [adapted from Vainshtein (1994)] (p. 42) | html | pdf |
    - Fig. 1.4.2.1. General-position and symmetry-operations blocks for the space group , No (p. 50) | html | pdf |
    - Fig. 1.4.2.2. General-position and symmetry-operations blocks as given in the space-group tables for space group Fmm2 (42) (p. 51) | html | pdf |
    - Fig. 1.4.2.3. Symmetry-element diagram for space group Fmm2 (42) (orthogonal projection along [001]) (p. 56) | html | pdf |
    - Fig. 1.4.2.4. General-position and symmetry-operations blocks as given in the space-group tables for space group P4mm (99) (p. 56) | html | pdf |
    - Fig. 1.4.2.5. Symmetry-element diagram for space group P4mm (99) (orthogonal projection along [001]) (p. 56) | html | pdf |
    - Fig. 1.4.2.6. Symmetry-element diagram (left) and general-position diagram (right) for the space group , No (p. 57) | html | pdf |
    - Fig. 1.4.2.7. Symmetry-element diagram (left) and general-position diagram (right) for the space group P2, No (p. 58) | html | pdf |
    - Fig. 1.4.2.8. General-position diagrams for the space group (214) (p. 58) | html | pdf |
    - Fig. 1.4.4.1. General-position block as given in the space-group tables for space group P4bm (100) (p. 63) | html | pdf |
    - Fig. 1.4.4.2. Symmetry-element diagram for the space group P4bm (100) for the orthogonal projection along [001] (p. 64) | html | pdf |
    - Fig. 1.4.4.3. Symmetry-element diagram for the space group P4 (75) for the orthogonal projection along [001] (p. 65) | html | pdf |
    - Fig. 1.4.4.4. Symmetry-element diagrams for the space group (17) for orthogonal projections along [001], [010], [100] (left to right) (p. 65) | html | pdf |
    - Fig. 1.4.5.1. Duality between section and projection (p. 68) | html | pdf |
    - Fig. 1.4.5.2. Symmetry-element diagrams for the space group (31) for orthogonal projections along [100] (left), [010] (middle) and [001] (right) (p. 70) | html | pdf |
    - Fig. 1.4.5.3. Symmetry-element diagram for the layer group (32) (p. 70) | html | pdf |
    - Fig. 1.4.5.4. Orthogonal projection along [001] of the symmetry-element diagram for (31) (top) and the diagram for plane group p2mg (7) (bottom) (p. 72) | html | pdf |
    - Fig. 1.4.5.5. `Symmetry of special projections' block of $[P\bar{4}b2]$ (117) as given in the space-group tables (p. 73) | html | pdf |
    - Fig. 1.4.5.6. Orthogonal projection along [001] of the symmetry-element diagram for $[P\bar{4}b2]$ (117) (left) and the diagram for plane group p4gm (12) (right) (p. 73) | html | pdf |
  - Tables
    - Table 1.4.1.1. The structure of the Hermann–Mauguin symbols for the space groups (p. 45) | html | pdf |
    - Table 1.4.1.2. The structure of the Hermann–Mauguin symbols for the plane groups (p. 48) | html | pdf |
    - Table 1.4.1.3. List of geometric crystal classes in which the Schoenflies sequence separates space groups belonging to the same arithmetic crystal class (p. 49) | html | pdf |
    - Table 1.4.2.1. Linear parts R of the Seitz symbols $[\{{\bi R}|{\bi v}\}]$ for space-group symmetry operations of cubic, tetragonal, orthorhombic, monoclinic and triclinic crystal systems (p. 52) | html | pdf |
    - Table 1.4.2.2. Linear parts R of the Seitz symbols $[\{{\bi R}|{\bi v}\}]$ for space-group symmetry operations of hexagonal and trigonal crystal systems (p. 52) | html | pdf |
    - Table 1.4.2.3. Linear parts R of the Seitz symbols $[\{{\bi R}|{\bi v}\}]$ for symmetry operations of rhombohedral space groups (rhombohedral-axes setting) (p. 52) | html | pdf |
    - Table 1.4.2.4. Linear parts R of the Seitz symbols $[\{{\bi R}|{\bi v}\}]$ for plane-group symmetry operations of oblique, rectangular and square crystal systems (p. 53) | html | pdf |
    - Table 1.4.2.5. Linear parts R of the Seitz symbols $[\{{\bi R}|{\bi v}\}]$ for plane-group symmetry operations of the hexagonal crystal system (p. 53) | html | pdf |
    - Table 1.4.2.6. Right coset decomposition of space group , No. 14 (unique axis b, cell choice 1) with respect to the normal subgroup of translations $[{\cal T}]$ (p. 54) | html | pdf |
    - Table 1.4.3.1. Sequence of generators for the crystal classes (p. 59) | html | pdf |
    - Table 1.4.3.2. Generation of the space group $[P6_122\equiv D_6^2 ]$ (178) (p. 60) | html | pdf |
    - Table 1.4.5.1. Coset representatives of (31) relative to its translation subgroup (p. 69) | html | pdf |
    - Table 1.4.5.2. Coset representatives of $[P\bar{4}b2]$ (117) relative to its translation subgroup (p. 73) | html | pdf |
- 1.5. Transformations of coordinate systems (pp. 75-105) | html | pdf | chapter contents |
  H. Wondratschek, M. I. Aroyo, B. Souvignier and G. Chapuis
  - 1.5.1. Origin shift and change of the basis (pp. 75-83) | html | pdf |
    H. Wondratschek and M. I. Aroyo
    - 1.5.1.1. Origin shift (pp. 75-76) | html | pdf |
    - 1.5.1.2. Change of the basis (pp. 76-83) | html | pdf |
    - 1.5.1.3. General change of coordinate system (p. 83) | html | pdf |
  - 1.5.2. Transformations of crystallographic quantities under coordinate transformations (pp. 83-87) | html | pdf |
    H. Wondratschek and M. I. Aroyo
    - 1.5.2.1. Covariant and contravariant quantities (p. 83) | html | pdf |
    - 1.5.2.2. Metric tensors of direct and reciprocal lattices (p. 84) | html | pdf |
    - 1.5.2.3. Transformation of matrix–column pairs of symmetry operations (p. 84) | html | pdf |
    - 1.5.2.4. Augmented-matrix formalism (pp. 84-86) | html | pdf |
    - 1.5.2.5. Example: paraelectric-to-ferroelectric phase transition of GeTe (pp. 86-87) | html | pdf |
  - 1.5.3. Transformations between different space-group descriptions (pp. 87-90) | html | pdf |
    G. Chapuis, H. Wondratschek and M. I. Aroyo
    - 1.5.3.1. Space groups with more than one description in this volume (pp. 87-88) | html | pdf |
    - 1.5.3.2. Examples (pp. 88-90) | html | pdf |
      - 1.5.3.2.1. Transformations between different settings of P2₁/c (pp. 88-90) | html | pdf |
      - 1.5.3.2.2. Transformation between the two origin-choice settings of I4₁/amd (p. 90) | html | pdf |
  - 1.5.4. Synoptic tables of plane and space groups (pp. 91-106) | html | pdf |
    B. Souvignier, G. Chapuis and H. Wondratschek
    - 1.5.4.1. Additional symmetry operations and symmetry elements (pp. 91-95) | html | pdf |
      - 1.5.4.1.1. Determining the type of a symmetry operation (pp. 91-92) | html | pdf |
      - 1.5.4.1.2. Cosets without additional types of symmetry elements (p. 92) | html | pdf |
      - 1.5.4.1.3. Examples with additional types of symmetry elements (pp. 92-95) | html | pdf |
    - 1.5.4.2. Synoptic table of the plane groups (p. 95) | html | pdf |
    - 1.5.4.3. Synoptic table of the space groups (pp. 95-106) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 1.5.1.1. The coordinates of the points X (or Y) with respect to the old origin O are x (y), and with respect to the new origin they are $[{\bi x}']$ $[({\bi y}')]$ (p. 75) | html | pdf |
    - Fig. 1.5.1.2. Monoclinic centred lattice, projected along the unique axis (p. 80) | html | pdf |
    - Fig. 1.5.1.3. Body-centred cell I with a_I, b_I, c_I and a corresponding primitive cell P with a_P, b_P, c_P (p. 80) | html | pdf |
    - Fig. 1.5.1.4. Face-centred cell F with a_F, b_F, c_F and a corresponding primitive cell P with a_P, b_P, c_P (p. 80) | html | pdf |
    - Fig. 1.5.1.5. Tetragonal lattices, projected along $[[00\bar{1}]]$ (p. 81) | html | pdf |
    - Fig. 1.5.1.6. Unit cells in the rhombohedral lattice (p. 81) | html | pdf |
    - Fig. 1.5.1.7. Hexagonal lattice projected along $[[00\bar{1}]]$ (p. 82) | html | pdf |
    - Fig. 1.5.1.8. Hexagonal lattice projected along $[[00\bar{1}]]$ (p. 82) | html | pdf |
    - Fig. 1.5.1.9. Rhombohedral lattice with a triple hexagonal unit cell a, b, c in obverse setting (p. 82) | html | pdf |
    - Fig. 1.5.1.10. Rhombohedral lattice with primitive rhombohedral cell a_rh, b_rh, c_rh and the three centred monoclinic cells (p. 82) | html | pdf |
    - Fig. 1.5.2.1. Illustration of the transformation of symmetry operations $[({\bi W}, {\bi w})]$ , also called a `mapping of mappings' (p. 84) | html | pdf |
    - Fig. 1.5.3.1. Three possible cell choices for the monoclinic space group $[P2_{1}/c]$ (14) with unique axis b (p. 87) | html | pdf |
    - Fig. 1.5.3.2. Two possible origin choices for the orthorhombic space group Pban (50) (p. 88) | html | pdf |
    - Fig. 1.5.3.3. General-position diagram of the space group R3m (160) showing the relation between the hexagonal and rhombohedral axes in the obverse setting: $[{\bf a}_{\rm rh}]$ = $[\textstyle{{1}\over{3}}(2{\bf a}_{\rm hex}+{\bf b}_{\rm hex}+{\bf c}_{\rm hex})]$ , $[{\bf b}_{\rm rh}]$ = $[\textstyle{{1}\over{3}}(-{\bf a}_{\rm hex}+{\bf b}_{\rm hex}+{\bf c}_{\rm hex})]$ , $[{\bf c}_{\rm rh}]$ = $[\textstyle{{1}\over{3}}(-{\bf a}_{\rm hex}]$ $[-2{\bf b}_{\rm hex}+{\bf c}_{\rm hex})]$ (p. 88) | html | pdf |
    - Fig. 1.5.4.1. Symmetry-element diagram for space group P23 (195) (p. 95) | html | pdf |
  - Tables
    - Table 1.5.1.1. Selected 3 × 3 transformation matrices $[{\bi P}]$ and $[{\bi Q} = {\bi P}^{ -1}]$ (pp. 77-80) | html | pdf |
    - Table 1.5.3.1. Transformation of reflection-condition data for P12₁/c1 to P112₁/a (p. 89) | html | pdf |
    - Table 1.5.4.1. Additional symmetry operations and their locations if the translation vector t is inclined to the symmetry axis or symmetry plane (p. 93) | html | pdf |
    - Table 1.5.4.2. Additional symmetry operations due to a centring vector t and their locations (p. 94) | html | pdf |
    - Table 1.5.4.3. List of plane-group symbols (p. 96) | html | pdf |
    - Table 1.5.4.4. List of space-group symbols for various settings and cells (pp. 97-105) | html | pdf |
- 1.6. Methods of space-group determination (pp. 107-131) | html | pdf | chapter contents |
  U. Shmueli, H. D. Flack and J. C. H. Spence
  - 1.6.1. Overview (p. 107) | html | pdf |
  - 1.6.2. Symmetry determination from single-crystal studies (pp. 107-111) | html | pdf |
    U. Shmueli and H. D. Flack
    - 1.6.2.1. Symmetry information from the diffraction pattern (pp. 107-109) | html | pdf |
    - 1.6.2.2. Structure-factor statistics and crystal symmetry (pp. 109-110) | html | pdf |
    - 1.6.2.3. Symmetry information from the structure solution (p. 110) | html | pdf |
    - 1.6.2.4. Restrictions on space groups (p. 111) | html | pdf |
    - 1.6.2.5. Pitfalls in space-group determination (p. 111) | html | pdf |
  - 1.6.3. Theoretical background of reflection conditions (pp. 112-114) | html | pdf |
    U. Shmueli
  - 1.6.4. Tables of reflection conditions and possible space groups (p. 114) | html | pdf |
    H. D. Flack and U. Shmueli
    - 1.6.4.1. Introduction (p. 114) | html | pdf |
    - 1.6.4.2. Examples of the use of the tables (p. 114) | html | pdf |
  - 1.6.5. Specialized methods of space-group determination (pp. 114-128) | html | pdf |
    H. D. Flack
    - 1.6.5.1. Applications of resonant scattering to symmetry determination (pp. 114-126) | html | pdf |
      - 1.6.5.1.1. Introduction (pp. 114-125) | html | pdf |
      - 1.6.5.1.2. Status of centrosymmetry and resonant scattering (pp. 125-126) | html | pdf |
      - 1.6.5.1.3. Resolution of noncentrosymmetric ambiguities (p. 126) | html | pdf |
      - 1.6.5.1.4. Data evaluation after structure refinement (p. 126) | html | pdf |
    - 1.6.5.2. Space-group determination in macromolecular crystallography (pp. 126-127) | html | pdf |
    - 1.6.5.3. Space-group determination from powder diffraction (pp. 127-128) | html | pdf |
  - 1.6.6. Space groups for nanocrystals by electron microscopy (pp. 128-129) | html | pdf |
    J. C. H. Spence
  - References | html | pdf |
  - Figures
    - Fig. 1.6.2.1. Ideal p.d.f.'s for the equal-atom case (p. 109) | html | pdf |
    - Fig. 1.6.2.2. Exact p.d.f.'s (p. 110) | html | pdf |
    - Fig. 1.6.5.1. Data-evaluation plot for crystal Ex2 (p. 127) | html | pdf |
    - Fig. 1.6.5.2. Data-evaluation plot for crystal Ex1 (p. 127) | html | pdf |
    - Fig. 1.6.6.1. Polarity determination by convergent-beam electron diffraction (p. 129) | html | pdf |
  - Tables
    - Table 1.6.2.1. The ability of the procedures described in Sections 1.6.2.1 and 1.6.5.1 to distinguish between space groups (p. 108) | html | pdf |
    - Table 1.6.2.2. The numerical values of several low-order moments of , based on equation (1.6.2.3) (p. 109) | html | pdf |
    - Table 1.6.3.1. Effect of lattice type on conditions for possible reflections (p. 113) | html | pdf |
    - Table 1.6.3.2. Effect of some glide reflections on conditions for possible reflections (p. 113) | html | pdf |
    - Table 1.6.3.3. Effect of some screw rotations on conditions for possible reflections (p. 113) | html | pdf |
    - Table 1.6.4.1. Summary of Tables 1.6.4.2–1.6.4.30 (p. 115) | html | pdf |
    - Table 1.6.4.2. Reflection conditions and possible space groups with Bravais lattice aP and Laue class $[\overline{1}]$ ; Patterson symmetry $[P\overline{1}]$ (p. 115) | html | pdf |
    - Table 1.6.4.3. Reflection conditions and possible space groups with Bravais lattice and Laue class 2/m; (monoclinic, unique axis b); Patterson symmetry (p. 115) | html | pdf |
    - Table 1.6.4.4. Reflection conditions and possible space groups with Bravais lattice mS (mC, mA, mI) and Laue class 2/m (monoclinic, unique axis b); Patterson symmetry , , (p. 116) | html | pdf |
    - Table 1.6.4.5. Reflection conditions and possible space groups with Bravais lattice mP and Laue class 2/m (monoclinic, unique axis c); Patterson symmetry (p. 116) | html | pdf |
    - Table 1.6.4.6. Reflection conditions and possible space groups with Bravais lattice mS (mA, mB, mI) and Laue class 2/m (monoclinic, unique axis c); Patterson symmetry , , (p. 116) | html | pdf |
    - Table 1.6.4.7. Reflection conditions and possible space groups with Bravais lattice oP and Laue class mmm; Patterson symmetry (pp. 117-118) | html | pdf |
    - Table 1.6.4.8. Reflection conditions and possible space groups with Bravais lattice oS (oC setting) and Laue class mmm; Patterson symmetry Cmmm (p. 119) | html | pdf |
    - Table 1.6.4.9. Reflection conditions and possible space groups with Bravais lattice oS (oB setting) and Laue class mmm; Patterson symmetry Bmmm (p. 119) | html | pdf |
    - Table 1.6.4.10. Reflection conditions and possible space groups with Bravais lattice oS (oA setting) and Laue class mmm; Patterson symmetry Ammm (p. 119) | html | pdf |
    - Table 1.6.4.11. Reflection conditions and possible space groups with Bravais lattice oI and Laue class mmm; Patterson symmetry Immm (p. 120) | html | pdf |
    - Table 1.6.4.12. Reflection conditions and possible space groups with Bravais lattice oF and Laue class mmm; Patterson symmetry Fmmm (p. 120) | html | pdf |
    - Table 1.6.4.13. Reflection conditions and possible space groups with Bravais lattice tP and Laue class 4/m; hk are permutable; Patterson symmetry P4/m (p. 120) | html | pdf |
    - Table 1.6.4.14. Reflection conditions and possible space groups with Bravais lattice tP and Laue class 4/mmm; hk are permutable; Patterson symmetry P4/mmm (p. 121) | html | pdf |
    - Table 1.6.4.15. Reflection conditions and possible space groups with Bravais lattice tI and Laue class 4/m; hk are permutable; Patterson symmetry I4/m (p. 121) | html | pdf |
    - Table 1.6.4.16. Reflection conditions and possible space groups with Bravais lattice tI and Laue class 4/mmm; hk are permutable; Patterson symmetry I4/mmm (p. 122) | html | pdf |
    - Table 1.6.4.17. Reflection conditions and possible space groups with Bravais lattice hP and Laue class $[\overline{3}]$ ; hki are permutable; Patterson symmetry $[P\overline{3}]$ (p. 122) | html | pdf |
    - Table 1.6.4.18. Reflection conditions and possible space groups with Bravais lattice hP and Laue classes $[\overline{3}1m]$ and $[\overline{3}m1]$ ; hki are permutable; Patterson symmetry $[P\overline{3}1m]$ and $[P\overline{3}m1]$ (p. 122) | html | pdf |
    - Table 1.6.4.19. Reflection conditions and possible space groups with Bravais lattice hP and Laue class 6/m; hki are permutable; Patterson symmetry P6/m (p. 122) | html | pdf |
    - Table 1.6.4.20. Reflection conditions and possible space groups with Bravais lattice hP and Laue class 6/mmm; hki are permutable; Patterson symmetry P6/mmm (p. 123) | html | pdf |
    - Table 1.6.4.21. Reflection conditions and possible space groups with Bravais lattice hR and Laue class $[\overline{3}]$ (hexagonal axes); hki are permutable; Patterson symmetry $[R\overline{3}]$ ; Ov = obverse setting; Rv = reverse setting (p. 123) | html | pdf |
    - Table 1.6.4.22. Reflection conditions and possible space groups with Bravais lattice hR and Laue class $[\overline{3}m]$ (hexagonal axes); hki are permutable; Patterson symmetry $[R\overline{3}m]$ ; Ov = obverse setting; Rv = reverse setting (p. 123) | html | pdf |
    - Table 1.6.4.23. Reflection conditions and possible space groups with Bravais lattice hR and Laue class $[\overline{3}]$ (rhombohedral axes); hkl are permutable; Patterson symmetry $[R\overline{3}]$ (p. 123) | html | pdf |
    - Table 1.6.4.24. Reflection conditions and possible space groups with Bravais lattice hR and Laue class $[\overline{3}m]$ (rhombohedral axes); hkl are permutable; Patterson symmetry $[R\overline{3}m]$ (p. 123) | html | pdf |
    - Table 1.6.4.25. Reflection conditions and possible space groups with Bravais lattice cP and Laue class $[m\overline{3}]$ ; hkl are cyclically permutable; Patterson symmetry $[Pm\overline{3}]$ (p. 124) | html | pdf |
    - Table 1.6.4.26. Reflection conditions and possible space groups with Bravais lattice cP and Laue class $[m\overline{3}m]$ ; hkl are permutable; Patterson symmetry $[Pm\overline{3}m]$ (p. 124) | html | pdf |
    - Table 1.6.4.27. Reflection conditions and possible space groups with Bravais lattice cI and Laue class $[m\overline{3}]$ ; hkl are cyclically permutable; Patterson symmetry $[Im\overline{3}]$ (p. 124) | html | pdf |
    - Table 1.6.4.28. Reflection conditions and possible space groups with Bravais lattice cI and Laue class $[m\overline{3}m]$ ; hkl are permutable; Patterson symmetry $[Im\overline{3}m]$ (p. 124) | html | pdf |
    - Table 1.6.4.29. Reflection conditions and possible space groups with Bravais lattice cF and Laue class $[m\overline{3}]$ ; hkl are cyclically permutable; Patterson symmetry $[Fm\overline{3}]$ (p. 124) | html | pdf |
    - Table 1.6.4.30. Reflection conditions and possible space groups with Bravais lattice cF and Laue class $[m\overline{3}m]$ ; hkl are permutable; Patterson symmetry $[Fm\overline{3}m]$ (p. 125) | html | pdf |
    - Table 1.6.5.1. R_merge values for Ex2 for the 589 sets of general reflections of mmm which have all eight measurements in the set (p. 126) | html | pdf |
    - Table 1.6.5.2. R_merge values for Ex1 for the 724 sets of general reflections of 2/m which have all four measurements in the set (p. 126) | html | pdf |
- 1.7. Topics on space groups treated in Volumes A1 and E of International Tables for Crystallography (pp. 132-139) | html | pdf | chapter contents |
  H. Wondratschek, U. Müller, D. B. Litvin and V. Kopský
  - 1.7.1. Subgroups and supergroups of space groups (pp. 132-134) | html | pdf |
    H. Wondratschek
    - 1.7.1.1. Translationengleiche (or t-) subgroups of space groups (p. 133) | html | pdf |
    - 1.7.1.2. Klassengleiche (or k-) subgroups of space groups (p. 134) | html | pdf |
    - 1.7.1.3. Isomorphic subgroups of space groups (p. 134) | html | pdf |
    - 1.7.1.4. Supergroups (p. 134) | html | pdf |
  - 1.7.2. Relations between Wyckoff positions for group–subgroup-related space groups (pp. 135-136) | html | pdf |
    U. Müller
    - 1.7.2.1. Symmetry relations between crystal structures (p. 135) | html | pdf |
    - 1.7.2.2. Substitution derivatives (p. 135) | html | pdf |
    - 1.7.2.3. Phase transitions (pp. 135-136) | html | pdf |
    - 1.7.2.4. Domain structures (p. 136) | html | pdf |
    - 1.7.2.5. Presentation of the relations between the Wyckoff positions among group–subgroup-related space groups (p. 136) | html | pdf |
  - 1.7.3. Relationships between space groups and subperiodic groups (pp. 136-139) | html | pdf |
    D. B. Litvin and V. Kopský
    - 1.7.3.1. Layer symmetries in three-dimensional crystal structures (pp. 137-138) | html | pdf |
    - 1.7.3.2. The symmetry of domain walls (pp. 138-139) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 1.7.1.1. Space group with t-subgroups of index 2 and 4 (p. 133) | html | pdf |
    - Fig. 1.7.2.1. Group–subgroup relation from the aristotype diamond to its hettotype zinc blende (p. 135) | html | pdf |
    - Fig. 1.7.2.2. Group–subgroup relations (Bärnighausen tree) from the ReO₃ type to two polymorphic forms of WO₃ (p. 136) | html | pdf |
    - Fig. 1.7.3.1. The scanning table for the space-group type $[P\overline 3 m1]$ (164) and orientation orbit (0001), and the structure of cadmium iodide, CdI₂ (p. 137) | html | pdf |
    - Fig. 1.7.3.2. At the centre is the structure of the cubic phase of barium titanate, BaTiO₃, of symmetry type $[Pm\overline 3 m]$ , surrounded by the structures of four of the six single-domain states of the tetragonal phase of symmetry type P4mm (p. 138) | html | pdf |
    - Fig. 1.7.3.3. The domain pair of symmetry P4_z/m_zm_xm_xy consisting of the superposition of those two single-domain states of tetragonal symmetry P4_zm_xm_xy shown in Fig. 1.7.3.2 (p. 138) | html | pdf |
    - Fig. 1.7.3.4. The domain twin consisting of the two domain states of tetragonal symmetry P4_zm_xm_xy and a domain wall of orientation (010) passing through the origin of the domain twin (p. 139) | html | pdf |

The space-group tables
- 2.1. Guide to the use of the space-group tables (pp. 142-174) | html | pdf | chapter contents |
  Th. Hahn, A. Looijenga-Vos, M. I. Aroyo, H. D. Flack, K. Momma and P. Konstantinov
  - 2.1.1. Conventional descriptions of plane and space groups (pp. 142-144) | html | pdf |
    Th. Hahn and A. Looijenga-Vos
    - 2.1.1.1. Classification of space groups (p. 142) | html | pdf |
    - 2.1.1.2. Conventional coordinate systems and cells (pp. 142-144) | html | pdf |
  - 2.1.2. Symbols of symmetry elements (pp. 144-148) | html | pdf |
    Th. Hahn and M. I. Aroyo
  - 2.1.3. Contents and arrangement of the tables (pp. 150-172) | html | pdf |
    Th. Hahn and A. Looijenga-Vos
    - 2.1.3.1. General layout (p. 150) | html | pdf |
    - 2.1.3.2. Space groups with more than one description (p. 150) | html | pdf |
    - 2.1.3.3. Headline (p. 151) | html | pdf |
    - 2.1.3.4. International (Hermann–Mauguin) symbols for plane groups and space groups (pp. 151-152) | html | pdf |
    - 2.1.3.5. Patterson symmetry (pp. 152-154) | html | pdf |
      H. D. Flack
    - 2.1.3.6. Space-group diagrams (pp. 154-158) | html | pdf |
      - 2.1.3.6.1. Plane groups (p. 154) | html | pdf |
      - 2.1.3.6.2. Triclinic space groups (pp. 154-155) | html | pdf |
      - 2.1.3.6.3. Monoclinic space groups (cf. Sections 2.1.3.2 and 2.1.3.15) (p. 155) | html | pdf |
      - 2.1.3.6.4. Orthorhombic space groups and orthorhombic settings (pp. 155-157) | html | pdf |
      - 2.1.3.6.5. Tetragonal, trigonal P and hexagonal P space groups (p. 157) | html | pdf |
      - 2.1.3.6.6. Trigonal R (rhombohedral) space groups (p. 157) | html | pdf |
      - 2.1.3.6.7. Cubic space groups (p. 157) | html | pdf |
      - 2.1.3.6.8. Diagrams of the general position (by K. Momma and M. I. Aroyo) (p. 158) | html | pdf |
    - 2.1.3.7. Origin (pp. 158-159) | html | pdf |
    - 2.1.3.8. Asymmetric unit (pp. 159-160) | html | pdf |
    - 2.1.3.9. Symmetry operations (pp. 160-161) | html | pdf |
    - 2.1.3.10. Generators (pp. 161-162) | html | pdf |
    - 2.1.3.11. Positions (p. 162) | html | pdf |
    - 2.1.3.12. Oriented site-symmetry symbols (p. 163) | html | pdf |
    - 2.1.3.13. Reflection conditions (pp. 163-167) | html | pdf |
    - 2.1.3.14. Symmetry of special projections (pp. 167-169) | html | pdf |
    - 2.1.3.15. Monoclinic space groups (pp. 169-172) | html | pdf |
    - 2.1.3.16. Crystallographic groups in one dimension (p. 172) | html | pdf |
  - 2.1.4. Computer production of the space-group tables (pp. 172-173) | html | pdf |
    P. Konstantinov and K. Momma
  - References | html | pdf |
  - Figures
    - Fig. 2.1.3.1. Triclinic space groups ( $[\hbox{\sf G}]$ = general-position diagram) (p. 154) | html | pdf |
    - Fig. 2.1.3.2. Monoclinic space groups, setting with unique axis b ( $[\hbox{\sf G}]$ = general-position diagram) (p. 155) | html | pdf |
    - Fig. 2.1.3.3. Monoclinic space groups, setting with unique axis c ( $[\hbox{\sf G}]$ = general-position diagram) (p. 155) | html | pdf |
    - Fig. 2.1.3.4. Monoclinic space groups, cell choices 1, 2, 3 (p. 155) | html | pdf |
    - Fig. 2.1.3.5. Orthorhombic space groups. Diagrams for the `standard setting' as described in the space-group tables ( $[\hbox{\sf G}]$ = general-position diagram) (p. 155) | html | pdf |
    - Fig. 2.1.3.6. Orthorhombic space groups. The three projections of the symmetry elements with the six setting symbols (see text) (p. 156) | html | pdf |
    - Fig. 2.1.3.7. Tetragonal space groups ( $[\hbox{\sf G}]$ = general-position diagram) (p. 157) | html | pdf |
    - Fig. 2.1.3.8. Trigonal P and hexagonal P space groups ( $[\hbox{\sf G}]$ = general-position diagram) (p. 157) | html | pdf |
    - Fig. 2.1.3.9. Rhombohedral space groups (p. 157) | html | pdf |
    - Fig. 2.1.3.10. Cubic space groups (p. 157) | html | pdf |
    - Fig. 2.1.3.11. Boundary planes of asymmetric units occurring in the space-group tables (p. 160) | html | pdf |
    - Fig. 2.1.3.12. The three primitive two-dimensional cells which are spanned by the shortest three translation vectors e, f, g in the monoclinic plane (p. 170) | html | pdf |
    - Fig. 2.1.3.13. The two line groups (one-dimensional space groups) (p. 172) | html | pdf |
  - Tables
    - Table 2.1.1.1. Crystal families, crystal systems, conventional coordinate systems and Bravais lattices in one, two and three dimensions (p. 143) | html | pdf |
    - Table 2.1.1.2. Symbols for the conventional centring types of one-, two- and three-dimensional cells (p. 144) | html | pdf |
    - Table 2.1.2.1. Symbols for symmetry elements and for the corresponding symmetry operations in one, two and three dimensions (p. 145) | html | pdf |
    - Table 2.1.2.2. Graphical symbols of symmetry planes normal to the plane of projection (three dimensions) and symmetry lines in the plane of the figure (two dimensions) (p. 146) | html | pdf |
    - Table 2.1.2.3. Graphical symbols of symmetry planes parallel to the plane of projection (p. 146) | html | pdf |
    - Table 2.1.2.4. Graphical symbols of symmetry planes inclined to the plane of projection (in cubic space groups of classes $[\overline{4}{3m}]$ and $[m\overline{3}m]$ only) (p. 147) | html | pdf |
    - Table 2.1.2.5. Graphical symbols of symmetry axes normal to the plane of projection and symmetry points in the plane of the figure (p. 148) | html | pdf |
    - Table 2.1.2.6. Graphical symbols of symmetry axes parallel to the plane of projection (p. 149) | html | pdf |
    - Table 2.1.2.7. Graphical symbols of symmetry axes inclined to the plane of projection (in cubic space groups only) (p. 149) | html | pdf |
    - Table 2.1.3.1. Lattice symmetry directions for two and three dimensions (p. 151) | html | pdf |
    - Table 2.1.3.2. Changes in Hermann–Mauguin symbols for two-dimensional groups (p. 152) | html | pdf |
    - Table 2.1.3.3. Patterson symmetries and symmetries of Patterson functions for space groups and plane groups (pp. 153-154) | html | pdf |
    - Table 2.1.3.4. Numbers of distinct projections and different Hermann–Mauguin symbols for the orthorhombic space groups (p. 156) | html | pdf |
    - Table 2.1.3.5. Examples of origin statements (p. 159) | html | pdf |
    - Table 2.1.3.6. Integral reflection conditions for centred cells (lattices) (p. 164) | html | pdf |
    - Table 2.1.3.7. Zonal and serial reflection conditions for glide planes and screw axes (cf. Table 2.1.2.1) (pp. 165-166) | html | pdf |
    - Table 2.1.3.8. Reflection conditions for the plane groups (p. 166) | html | pdf |
    - Table 2.1.3.9. Cell parameters a′, b′, γ′ of the two-dimensional cell in terms of cell parameters a, b, c, α, β, γ of the three-dimensional cell for the projections listed in the space-group tables of Chapter 2.3 (p. 168) | html | pdf |
    - Table 2.1.3.10. Projections of crystallographic symmetry elements (p. 169) | html | pdf |
    - Table 2.1.3.11. Monoclinic setting symbols (p. 170) | html | pdf |
    - Table 2.1.3.12. Symbols for centring types and glide planes of monoclinic space groups (p. 171) | html | pdf |
- 2.2. The 17 plane groups (two-dimensional space groups) (pp. 175-192) | html | | chapter contents |
  - Plane group 1, p1 (p. 176) | html | pdf |
  - Plane group 2, p2 (p. 177) | html | pdf |
  - Plane group 3, pm (p. 178) | html | pdf |
  - Plane group 4, pg (p. 179) | html | pdf |
  - Plane group 5, cm (p. 180) | html | pdf |
  - Plane group 6, p2mm (p. 181) | html | pdf |
  - Plane group 7, p2mg (p. 182) | html | pdf |
  - Plane group 8, p2gg (p. 183) | html | pdf |
  - Plane group 9, c2mm (p. 184) | html | pdf |
  - Plane group 10, p4 (p. 185) | html | pdf |
  - Plane group 11, p4mm (p. 186) | html | pdf |
  - Plane group 12, p4gm (p. 187) | html | pdf |
  - Plane group 13, p3 (p. 188) | html | pdf |
  - Plane group 14, p3m1 (p. 189) | html | pdf |
  - Plane group 15, p31m (p. 190) | html | pdf |
  - Plane group 16, p6 (p. 191) | html | pdf |
  - Plane group 17, p6mm (p. 192) | html | pdf |
- 2.3. The 230 space groups (pp. 193-687) | html | | chapter contents |
  - Space group 1, P1 (p. 194) | html | pdf |
  - Space group 2, P-1 (p. 195) | html | pdf |
  - Space group 3, P2 (pp. 196-197) | html | pdf |
    - Space group 3, unique axis b, P2 (p. 196) | html | pdf |
    - Space group 3, unique axis c, P2 (p. 197) | html | pdf |
  - Space group 4, P2₁ (pp. 198-199) | html | pdf |
    - Space group 4, unique axis b, P2₁ (p. 198) | html | pdf |
    - Space group 4, unique axis c, P2₁ (p. 199) | html | pdf |
  - Space group 5, C2 (pp. 200-207) | html | pdf |
    - Space group 5, unique axis b, C2 (pp. 200-203) | html | pdf |
    - Space group 5, unique axis c, C2 (pp. 204-207) | html | pdf |
  - Space group 6, Pm (pp. 208-209) | html | pdf |
    - Space group 6, unique axis b, Pm (p. 208) | html | pdf |
    - Space group 6, unique axis c, Pm (p. 209) | html | pdf |
  - Space group 7, Pc (pp. 210-215) | html | pdf |
    - Space group 7, unique axis b, Pc (pp. 210-212) | html | pdf |
    - Space group 7, unique axis c, Pc (pp. 213-215) | html | pdf |
  - Space group 8, Cm (pp. 216-223) | html | pdf |
    - Space group 8, unique axis b, Cm (pp. 216-219) | html | pdf |
    - Space group 8, unique axis c, Cm (pp. 220-223) | html | pdf |
  - Space group 9, Cc (pp. 224-229) | html | pdf |
    - Space group 9, unique axis b, Cc (pp. 224-226) | html | pdf |
    - Space group 9, unique axis c, Cc (pp. 227-229) | html | pdf |
  - Space group 10, P2/m (pp. 230-233) | html | pdf |
    - Space group 10, unique axis b, P2/m (pp. 230-231) | html | pdf |
    - Space group 10, unique axis c, P2/m (pp. 232-233) | html | pdf |
  - Space group 11, P2₁/m (pp. 234-235) | html | pdf |
    - Space group 11, unique axis b, P2₁/m (p. 234) | html | pdf |
    - Space group 11, unique axis c, P2₁/m (p. 235) | html | pdf |
  - Space group 12, C2/m (pp. 236-243) | html | pdf |
    - Space group 12, unique axis b, C2/m (pp. 236-239) | html | pdf |
    - Space group 12, unique axis c, C2/m (pp. 240-243) | html | pdf |
  - Space group 13, P2/c (pp. 244-251) | html | pdf |
    - Space group 13, unique axis b, P2/c (pp. 244-247) | html | pdf |
    - Space group 13, unique axis c, P2/c (pp. 248-251) | html | pdf |
  - Space group 14, P2₁/c (pp. 252-259) | html | pdf |
    - Space group 14, unique axis b, P2₁/c (pp. 252-255) | html | pdf |
    - Space group 14, unique axis c, P2₁/c (pp. 256-259) | html | pdf |
  - Space group 15, C2/c (pp. 260-267) | html | pdf |
    - Space group 15, unique axis b, C2/c (pp. 260-263) | html | pdf |
    - Space group 15, unique axis c, C2/c (pp. 264-267) | html | pdf |
  - Space group 16, P222 (pp. 268-269) | html | pdf |
  - Space group 17, P222₁ (pp. 270-271) | html | pdf |
  - Space group 18, P2₁2₁2 (p. 272) | html | pdf |
  - Space group 19, P2₁2₁2₁ (p. 273) | html | pdf |
  - Space group 20, C222₁ (pp. 274-275) | html | pdf |
  - Space group 21, C222 (pp. 276-277) | html | pdf |
  - Space group 22, F222 (pp. 278-279) | html | pdf |
  - Space group 23, I222 (pp. 280-281) | html | pdf |
  - Space group 24, I2₁2₁2₁ (pp. 282-283) | html | pdf |
  - Space group 25, Pmm2 (p. 284) | html | pdf |
  - Space group 26, Pmc2₁ (p. 285) | html | pdf |
  - Space group 27, Pcc2 (p. 286) | html | pdf |
  - Space group 28, Pma2 (p. 287) | html | pdf |
  - Space group 29, Pca2₁ (p. 288) | html | pdf |
  - Space group 30, Pnc2 (p. 289) | html | pdf |
  - Space group 31, Pmn2₁ (p. 290) | html | pdf |
  - Space group 32, Pba2 (p. 291) | html | pdf |
  - Space group 33, Pna2₁ (p. 292) | html | pdf |
  - Space group 34, Pnn2 (p. 293) | html | pdf |
  - Space group 35, Cmm2 (pp. 294-295) | html | pdf |
  - Space group 36, Cmc2₁ (pp. 296-297) | html | pdf |
  - Space group 37, Ccc2 (pp. 298-299) | html | pdf |
  - Space group 38, Amm2 (pp. 300-301) | html | pdf |
  - Space group 39, Aem2 (pp. 302-303) | html | pdf |
  - Space group 40, Ama2 (pp. 304-305) | html | pdf |
  - Space group 41, Aea2 (pp. 306-307) | html | pdf |
  - Space group 42, Fmm2 (pp. 308-309) | html | pdf |
  - Space group 43, Fdd2 (pp. 310-311) | html | pdf |
  - Space group 44, Imm2 (pp. 312-313) | html | pdf |
  - Space group 45, Iba2 (pp. 314-315) | html | pdf |
  - Space group 46, Ima2 (pp. 316-317) | html | pdf |
  - Space group 47, Pmmm (pp. 318-319) | html | pdf |
  - Space group 48, Pnnn (pp. 320-323) | html | pdf |
    - Space group 48, origin choice 1, Pnnn (pp. 320-321) | html | pdf |
    - Space group 48, origin choice 2, Pnnn (pp. 322-323) | html | pdf |
  - Space group 49, Pccm (pp. 324-325) | html | pdf |
  - Space group 50, Pban (pp. 326-329) | html | pdf |
    - Space group 50, origin choice 1, Pban (pp. 326-327) | html | pdf |
    - Space group 50, origin choice 2, Pban (pp. 328-329) | html | pdf |
  - Space group 51, Pmma (pp. 330-331) | html | pdf |
  - Space group 52, Pnna (pp. 332-333) | html | pdf |
  - Space group 53, Pmna (pp. 334-335) | html | pdf |
  - Space group 54, Pcca (pp. 336-337) | html | pdf |
  - Space group 55, Pbam (pp. 338-339) | html | pdf |
  - Space group 56, Pccn (pp. 340-341) | html | pdf |
  - Space group 57, Pbcm (pp. 342-343) | html | pdf |
  - Space group 58, Pnnm (pp. 344-345) | html | pdf |
  - Space group 59, Pmmn (pp. 346-349) | html | pdf |
    - Space group 59, origin choice 1, Pmmn (pp. 346-347) | html | pdf |
    - Space group 59, origin choice 2, Pmmn (pp. 348-349) | html | pdf |
  - Space group 60, Pbcn (pp. 350-351) | html | pdf |
  - Space group 61, Pbca (pp. 352-353) | html | pdf |
  - Space group 62, Pnma (pp. 354-355) | html | pdf |
  - Space group 63, Cmcm (pp. 356-357) | html | pdf |
  - Space group 64, Cmce (pp. 358-359) | html | pdf |
  - Space group 65, Cmmm (pp. 360-361) | html | pdf |
  - Space group 66, Cccm (pp. 362-363) | html | pdf |
  - Space group 67, Cmme (pp. 364-365) | html | pdf |
  - Space group 68, Ccce (pp. 366-369) | html | pdf |
    - Space group 68, origin choice 1, Ccce (pp. 366-367) | html | pdf |
    - Space group 68, origin choice 2, Ccce (pp. 368-369) | html | pdf |
  - Space group 69, Fmmm (pp. 370-371) | html | pdf |
  - Space group 70, Fddd (pp. 372-375) | html | pdf |
    - Space group 70, origin choice 1, Fddd (pp. 372-373) | html | pdf |
    - Space group 70, origin choice 2, Fddd (pp. 374-375) | html | pdf |
  - Space group 71, Immm (pp. 376-377) | html | pdf |
  - Space group 72, Ibam (pp. 378-379) | html | pdf |
  - Space group 73, Ibca (pp. 380-381) | html | pdf |
  - Space group 74, Imma (pp. 382-383) | html | pdf |
  - Space group 75, P4 (p. 384) | html | pdf |
  - Space group 76, P4₁ (p. 385) | html | pdf |
  - Space group 77, P4₂ (p. 386) | html | pdf |
  - Space group 78, P4₃ (p. 387) | html | pdf |
  - Space group 79, I4 (p. 388) | html | pdf |
  - Space group 80, I4₁ (p. 389) | html | pdf |
  - Space group 81, P-4 (p. 390) | html | pdf |
  - Space group 82, I-4 (p. 391) | html | pdf |
  - Space group 83, P4/m (p. 392) | html | pdf |
  - Space group 84, P4₂/m (p. 393) | html | pdf |
  - Space group 85, P4/n (pp. 394-395) | html | pdf |
    - Space group 85, origin choice 1, P4/n (p. 394) | html | pdf |
    - Space group 85, origin choice 2, P4/n (p. 395) | html | pdf |
  - Space group 86, P4₂/n (pp. 396-397) | html | pdf |
    - Space group 86, origin choice 1, P4₂/n (p. 396) | html | pdf |
    - Space group 86, origin choice 2, P4₂/n (p. 397) | html | pdf |
  - Space group 87, I4/m (pp. 398-399) | html | pdf |
  - Space group 88, I4₁/a (pp. 400-401) | html | pdf |
    - Space group 88, origin choice 1, I4₁/a (p. 400) | html | pdf |
    - Space group 88, origin choice 2, I4₁/a (p. 401) | html | pdf |
  - Space group 89, P422 (pp. 402-403) | html | pdf |
  - Space group 90, P42₁2 (pp. 404-405) | html | pdf |
  - Space group 91, P4₁22 (p. 406) | html | pdf |
  - Space group 92, P4₁2₁2 (p. 407) | html | pdf |
  - Space group 93, P4₂22 (pp. 408-409) | html | pdf |
  - Space group 94, P4₂2₁2 (pp. 410-411) | html | pdf |
  - Space group 95, P4₃22 (p. 412) | html | pdf |
  - Space group 96, P4₃2₁2 (p. 413) | html | pdf |
  - Space group 97, I422 (pp. 414-415) | html | pdf |
  - Space group 98, I4₁22 (p. 416) | html | pdf |
  - Space group 99, P4mm (p. 417) | html | pdf |
  - Space group 100, P4bm (p. 418) | html | pdf |
  - Space group 101, P4₂cm (p. 419) | html | pdf |
  - Space group 102, P4₂nm (p. 420) | html | pdf |
  - Space group 103, P4cc (p. 421) | html | pdf |
  - Space group 104, P4nc (p. 422) | html | pdf |
  - Space group 105, P4₂mc (p. 423) | html | pdf |
  - Space group 106, P4₂bc (p. 424) | html | pdf |
  - Space group 107, I4mm (p. 425) | html | pdf |
  - Space group 108, I4cm (p. 426) | html | pdf |
  - Space group 109, I4₁md (p. 427) | html | pdf |
  - Space group 110, I4₁cd (p. 428) | html | pdf |
  - Space group 111, P-42m (p. 429) | html | pdf |
  - Space group 112, P-42c (p. 430) | html | pdf |
  - Space group 113, P-42₁m (p. 431) | html | pdf |
  - Space group 114, P-42₁c (p. 432) | html | pdf |
  - Space group 115, P-4m2 (p. 433) | html | pdf |
  - Space group 116, P-4c2 (p. 434) | html | pdf |
  - Space group 117, P-4b2 (p. 435) | html | pdf |
  - Space group 118, P-4n2 (p. 436) | html | pdf |
  - Space group 119, I-4m2 (p. 437) | html | pdf |
  - Space group 120, I-4c2 (pp. 438-439) | html | pdf |
  - Space group 121, I-42m (p. 440) | html | pdf |
  - Space group 122, I-42d (p. 441) | html | pdf |
  - Space group 123, P4/mmm (pp. 442-443) | html | pdf |
  - Space group 124, P4/mcc (pp. 444-445) | html | pdf |
  - Space group 125, P4/nbm (pp. 446-449) | html | pdf |
    - Space group 125, origin choice 1, P4/nbm (pp. 446-447) | html | pdf |
    - Space group 125, origin choice 2, P4/nbm (pp. 448-449) | html | pdf |
  - Space group 126, P4/nnc (pp. 450-453) | html | pdf |
    - Space group 126, origin choice 1, P4/nnc (pp. 450-451) | html | pdf |
    - Space group 126, origin choice 2, P4/nnc (pp. 452-453) | html | pdf |
  - Space group 127, P4/mbm (pp. 454-455) | html | pdf |
  - Space group 128, P4/mnc (pp. 456-457) | html | pdf |
  - Space group 129, P4/nmm (pp. 458-461) | html | pdf |
    - Space group 129, origin choice 1, P4/nmm (pp. 458-459) | html | pdf |
    - Space group 129, origin choice 2, P4/nmm (pp. 460-461) | html | pdf |
  - Space group 130, P4/ncc (pp. 462-463) | html | pdf |
    - Space group 130, origin choice 1, P4/ncc (p. 462) | html | pdf |
    - Space group 130, origin choice 2, P4/ncc (p. 463) | html | pdf |
  - Space group 131, P4₂/mmc (pp. 464-465) | html | pdf |
  - Space group 132, P4₂/mcm (pp. 466-467) | html | pdf |
  - Space group 133, P4₂/nbc (pp. 468-471) | html | pdf |
    - Space group 133, origin choice 1, P4₂/nbc (pp. 468-469) | html | pdf |
    - Space group 133, origin choice 2, P4₂/nbc (pp. 470-471) | html | pdf |
  - Space group 134, P4₂/nnm (pp. 472-475) | html | pdf |
    - Space group 134, origin choice 1, P4₂/nnm (pp. 472-473) | html | pdf |
    - Space group 134, origin choice 2, P4₂/nnm (pp. 474-475) | html | pdf |
  - Space group 135, P4₂/mbc (pp. 476-477) | html | pdf |
  - Space group 136, P4₂/mnm (pp. 478-479) | html | pdf |
  - Space group 137, P4₂/nmc (pp. 480-481) | html | pdf |
    - Space group 137, origin choice 1, P4₂/nmc (p. 480) | html | pdf |
    - Space group 137, origin choice 2, P4₂/nmc (p. 481) | html | pdf |
  - Space group 138, P4₂/ncm (pp. 482-485) | html | pdf |
    - Space group 138, origin choice 1, P4₂/ncm (pp. 482-483) | html | pdf |
    - Space group 138, origin choice 2, P4₂/ncm (pp. 484-485) | html | pdf |
  - Space group 139, I4/mmm (pp. 486-487) | html | pdf |
  - Space group 140, I4/mcm (pp. 488-489) | html | pdf |
  - Space group 141, I4₁/amd (pp. 490-493) | html | pdf |
    - Space group 141, origin choice 1, I4₁/amd (pp. 490-491) | html | pdf |
    - Space group 141, origin choice 2, I4₁/amd (pp. 492-493) | html | pdf |
  - Space group 142, I4₁/acd (pp. 494-497) | html | pdf |
    - Space group 142, origin choice 1, I4₁/acd (pp. 494-495) | html | pdf |
    - Space group 142, origin choice 2, I4₁/acd (pp. 496-497) | html | pdf |
  - Space group 143, P3 (p. 498) | html | pdf |
  - Space group 144, P3₁ (p. 499) | html | pdf |
  - Space group 145, P3₂ (p. 500) | html | pdf |
  - Space group 146, R3 (pp. 501-502) | html | pdf |
    - Space group 146, hexagonal axes, R3 (p. 501) | html | pdf |
    - Space group 146, rhombohedral axes, R3 (p. 502) | html | pdf |
  - Space group 147, P-3 (p. 503) | html | pdf |
  - Space group 148, R-3 (pp. 504-507) | html | pdf |
    - Space group 148, hexagonal axes, R-3 (pp. 504-505) | html | pdf |
    - Space group 148, rhombohedral axes, R-3 (pp. 506-507) | html | pdf |
  - Space group 149, P312 (p. 508) | html | pdf |
  - Space group 150, P321 (p. 509) | html | pdf |
  - Space group 151, P3₁12 (p. 510) | html | pdf |
  - Space group 152, P3₁21 (p. 511) | html | pdf |
  - Space group 153, P3₂12 (p. 512) | html | pdf |
  - Space group 154, P3₂21 (p. 513) | html | pdf |
  - Space group 155, R32 (pp. 514-517) | html | pdf |
    - Space group 155, hexagonal axes, R32 (pp. 514-515) | html | pdf |
    - Space group 155, rhombohedral axes, R32 (pp. 516-517) | html | pdf |
  - Space group 156, P3m1 (p. 518) | html | pdf |
  - Space group 157, P31m (p. 519) | html | pdf |
  - Space group 158, P3c1 (p. 520) | html | pdf |
  - Space group 159, P31c (p. 521) | html | pdf |
  - Space group 160, R3m (pp. 522-525) | html | pdf |
    - Space group 160, hexagonal axes, R3m (pp. 522-523) | html | pdf |
    - Space group 160, rhombohedral axes, R3m (pp. 524-525) | html | pdf |
  - Space group 161, R3c (pp. 526-529) | html | pdf |
    - Space group 161, hexagonal axes, R3c (pp. 526-527) | html | pdf |
    - Space group 161, rhombohedral axes, R3c (pp. 528-529) | html | pdf |
  - Space group 162, P-31m (pp. 530-531) | html | pdf |
  - Space group 163, P-31c (pp. 532-533) | html | pdf |
  - Space group 164, P-3m1 (p. 534) | html | pdf |
  - Space group 165, P-3c1 (p. 535) | html | pdf |
  - Space group 166, R-3m (pp. 536-539) | html | pdf |
    - Space group 166, hexagonal axes, R-3m (pp. 536-537) | html | pdf |
    - Space group 166, rhombohedral axes, R-3m (pp. 538-539) | html | pdf |
  - Space group 167, R-3c (pp. 540-543) | html | pdf |
    - Space group 167, hexagonal axes, R-3c (pp. 540-541) | html | pdf |
    - Space group 167, rhombohedral axes, R-3c (pp. 542-543) | html | pdf |
  - Space group 168, P6 (p. 544) | html | pdf |
  - Space group 169, P6₁ (p. 545) | html | pdf |
  - Space group 170, P6₅ (p. 546) | html | pdf |
  - Space group 171, P6₂ (p. 547) | html | pdf |
  - Space group 172, P6₄ (p. 548) | html | pdf |
  - Space group 173, P6₃ (p. 549) | html | pdf |
  - Space group 174, P-6 (p. 550) | html | pdf |
  - Space group 175, P6/m (p. 551) | html | pdf |
  - Space group 176, P6₃/m (pp. 552-553) | html | pdf |
  - Space group 177, P622 (pp. 554-555) | html | pdf |
  - Space group 178, P6₁22 (p. 556) | html | pdf |
  - Space group 179, P6₅22 (p. 557) | html | pdf |
  - Space group 180, P6₂22 (pp. 558-559) | html | pdf |
  - Space group 181, P6₄22 (pp. 560-561) | html | pdf |
  - Space group 182, P6₃22 (pp. 562-563) | html | pdf |
  - Space group 183, P6mm (p. 564) | html | pdf |
  - Space group 184, P6cc (p. 565) | html | pdf |
  - Space group 185, P6₃cm (p. 566) | html | pdf |
  - Space group 186, P6₃mc (p. 567) | html | pdf |
  - Space group 187, P-6m2 (pp. 568-569) | html | pdf |
  - Space group 188, P-6c2 (pp. 570-571) | html | pdf |
  - Space group 189, P-62m (pp. 572-573) | html | pdf |
  - Space group 190, P-62c (pp. 574-575) | html | pdf |
  - Space group 191, P6/mmm (pp. 576-577) | html | pdf |
  - Space group 192, P6/mcc (pp. 578-579) | html | pdf |
  - Space group 193, P6₃/mcm (pp. 580-581) | html | pdf |
  - Space group 194, P6₃/mmc (pp. 582-583) | html | pdf |
  - Space group 195, P23 (pp. 584-585) | html | pdf |
  - Space group 196, F23 (pp. 586-587) | html | pdf |
  - Space group 197, I23 (pp. 588-589) | html | pdf |
  - Space group 198, P2₁3 (pp. 590-591) | html | pdf |
  - Space group 199, I2₁3 (pp. 592-593) | html | pdf |
  - Space group 200, Pm-3 (pp. 594-595) | html | pdf |
  - Space group 201, Pn-3 (pp. 596-599) | html | pdf |
    - Space group 201, origin choice 1, Pn-3 (pp. 596-597) | html | pdf |
    - Space group 201, origin choice 2, Pn-3 (pp. 598-599) | html | pdf |
  - Space group 202, Fm-3 (pp. 600-601) | html | pdf |
  - Space group 203, Fd-3 (pp. 602-605) | html | pdf |
    - Space group 203, origin choice 1, Fd-3 (pp. 602-603) | html | pdf |
    - Space group 203, origin choice 2, Fd-3 (pp. 604-605) | html | pdf |
  - Space group 204, Im-3 (pp. 606-607) | html | pdf |
  - Space group 205, Pa-3 (pp. 608-609) | html | pdf |
  - Space group 206, Ia-3 (pp. 610-611) | html | pdf |
  - Space group 207, P432 (pp. 612-613) | html | pdf |
  - Space group 208, P4₂32 (pp. 614-615) | html | pdf |
  - Space group 209, F432 (pp. 616-617) | html | pdf |
  - Space group 210, F4₁32 (pp. 618-619) | html | pdf |
  - Space group 211, I432 (pp. 620-621) | html | pdf |
  - Space group 212, P4₃32 (pp. 622-623) | html | pdf |
  - Space group 213, P4₁32 (pp. 624-625) | html | pdf |
  - Space group 214, I4₁32 (pp. 626-627) | html | pdf |
  - Space group 215, P-43m (pp. 628-629) | html | pdf |
  - Space group 216, F-43m (pp. 630-631) | html | pdf |
  - Space group 217, I-43m (pp. 632-633) | html | pdf |
  - Space group 218, P-43n (pp. 634-635) | html | pdf |
  - Space group 219, F-43c (pp. 636-637) | html | pdf |
  - Space group 220, I-43d (pp. 638-639) | html | pdf |
  - Space group 221, Pm-3m (pp. 640-643) | html | pdf |
  - Space group 222, Pn-3n (pp. 644-651) | html | pdf |
    - Space group 222, origin choice 1, Pn-3n (pp. 644-647) | html | pdf |
    - Space group 222, origin choice 2, Pn-3n (pp. 648-651) | html | pdf |
  - Space group 223, Pm-3n (pp. 652-655) | html | pdf |
  - Space group 224, Pn-3m (pp. 656-663) | html | pdf |
    - Space group 224, origin choice 1, Pn-3m (pp. 656-659) | html | pdf |
    - Space group 224, origin choice 2, Pn-3m (pp. 660-663) | html | pdf |
  - Space group 225, Fm-3m (pp. 664-667) | html | pdf |
  - Space group 226, Fm-3c (pp. 668-671) | html | pdf |
  - Space group 227, Fd-3m (pp. 672-679) | html | pdf |
    - Space group 227, origin choice 1, Fd-3m (pp. 672-675) | html | pdf |
    - Space group 227, origin choice 2, Fd-3m (pp. 676-679) | html | pdf |
  - Space group 228, Fd-3c (pp. 680-687) | html | pdf |
    - Space group 228, origin choice 1, Fd-3c (pp. 680-683) | html | pdf |
    - Space group 228, origin choice 2, Fd-3c (pp. 684-687) | html | pdf |
  - Space group 229, Im-3m (pp. 688-691) | html | pdf |
  - Space group 230, Ia-3d (pp. 692-695) | html | pdf |

Advanced topics on space-group symmetry
- 3.1. Crystal lattices (pp. 698-718) | html | pdf | chapter contents |
  H. Burzlaff, H. Grimmer, B. Gruber, P. M. de Wolff and H. Zimmermann
  - 3.1.1. Bases and lattices (pp. 698-700) | html | pdf |
    H. Burzlaff and H. Zimmermann
    - 3.1.1.1. Description and transformation of bases (p. 698) | html | pdf |
    - 3.1.1.2. Lattices (p. 698) | html | pdf |
    - 3.1.1.3. Topological properties of lattices (p. 698) | html | pdf |
    - 3.1.1.4. Special bases for lattices (pp. 698-699) | html | pdf |
    - 3.1.1.5. Remarks (pp. 699-700) | html | pdf |
  - 3.1.2. Bravais types of lattices and other classifications (pp. 700-708) | html | pdf |
    H. Burzlaff and H. Zimmermann
    - 3.1.2.1. Classifications (p. 700) | html | pdf |
    - 3.1.2.2. Description of Bravais types of lattices (pp. 700-701) | html | pdf |
    - 3.1.2.3. Delaunay reduction and standardization (pp. 701-707) | html | pdf |
    - 3.1.2.4. Example of Delaunay reduction and standardization of the basis (pp. 707-708) | html | pdf |
  - 3.1.3. Reduced bases (pp. 709-714) | html | pdf |
    P. M. de Wolff
    - 3.1.3.1. Introduction (p. 709) | html | pdf |
    - 3.1.3.2. Definition (p. 709) | html | pdf |
    - 3.1.3.3. Main conditions (pp. 709-710) | html | pdf |
    - 3.1.3.4. Special conditions (pp. 710-712) | html | pdf |
    - 3.1.3.5. Lattice characters (pp. 712-713) | html | pdf |
    - 3.1.3.6. Applications (pp. 713-714) | html | pdf |
  - 3.1.4. Further properties of lattices (pp. 714-718) | html | pdf |
    B. Gruber and H. Grimmer
    - 3.1.4.1. Further kinds of reduced cells (p. 714) | html | pdf |
    - 3.1.4.2. Topological characterization of lattice characters (pp. 714-715) | html | pdf |
    - 3.1.4.3. A finer division of lattices (p. 715) | html | pdf |
    - 3.1.4.4. Conventional cells (pp. 715-717) | html | pdf |
    - 3.1.4.5. Conventional characters (pp. 717-718) | html | pdf |
    - 3.1.4.6. Sublattices (p. 718) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 3.1.2.1. Conventional cells of the three-dimensional Bravais types of lattices (p. 700) | html | pdf |
    - Fig. 3.1.2.2. Delaunay reduction of Gruber's example (cf (p. 708) | html | pdf |
    - Fig. 3.1.3.1. The net of lattice points in the plane of the reduced basis vectors a and b; OBAD is a primitive mesh (p. 710) | html | pdf |
    - Fig. 3.1.3.2. The effect of the special conditions (p. 710) | html | pdf |
    - Fig. 3.1.3.3. The effect of the special conditions (p. 711) | html | pdf |
    - Fig. 3.1.3.4. The effect of the special conditions (p. 711) | html | pdf |
    - Fig. 3.1.3.5. The effect of the special conditions (p. 711) | html | pdf |
    - Fig. 3.1.4.1. A set M in $[{\bb E}^{2}]$ consisting of three components (p. 714) | html | pdf |
    - Fig. 3.1.4.2. A convex set in $[{\bb E}^{2}]$ (p. 715) | html | pdf |
    - Fig. 3.1.4.3. The Bravais-lattice type of the three-dimensional lattice at the upper end of a line is a limiting case of the type at the lower end (p. 717) | html | pdf |
    - Fig. 3.1.4.4. The Bravais-lattice type of the two-dimensional lattice at the upper end of a line is a limiting case of the type at the lower end (p. 717) | html | pdf |
    - Fig. 3.1.4.5. Three possible decompositions of a two-dimensional lattice L into sublattices of index 2 (p. 718) | html | pdf |
  - Tables
    - Table 3.1.1.1. Lattice point-group symmetries (p. 699) | html | pdf |
    - Table 3.1.2.1. Two-dimensional Bravais types of lattices (p. 701) | html | pdf |
    - Table 3.1.2.2. Three-dimensional Bravais types of lattices (pp. 702-703) | html | pdf |
    - Table 3.1.2.3. Delaunay types of lattices (`Symmetrische Sorten') (pp. 704-707) | html | pdf |
    - Table 3.1.2.4. Delaunay reduction for Gruber's example (p. 708) | html | pdf |
    - Table 3.1.2.5. Discussion of Gruber's example using the cell surface (p. 708) | html | pdf |
    - Table 3.1.3.1. The parameters $[D = {\bf b}\cdot {\bf c}]$ , $[E = {\bf a}\cdot {\bf c}]$ and $[F = {\bf a}\cdot {\bf b}]$ of the 44 lattice characters ( $[A = {\bf a}\cdot {\bf a},\ B = {\bf b}\cdot {\bf b},\ C = {\bf c}\cdot {\bf c}]$ ) (p. 712) | html | pdf |
    - Table 3.1.3.2. Lattice characters described by relations between conventional cell parameters (p. 713) | html | pdf |
    - Table 3.1.4.1. Conventional cells for the three-dimensional Bravais types of lattices and their limiting cases (p. 716) | html | pdf |
    - Table 3.1.4.2. Conventional cells for the five two-dimensional Bravais types of lattices and their limiting cases (p. 717) | html | pdf |
    - Table 3.1.4.3. Conventional characters (p. 718) | html | pdf |
- 3.2. Point groups and crystal classes (pp. 720-776) | html | pdf | chapter contents |
  Th. Hahn, H. Klapper, U. Müller and M. I. Aroyo
  - 3.2.1. Crystallographic and noncrystallographic point groups (pp. 720-737) | html | pdf |
    Th. Hahn and H. Klapper
    - 3.2.1.1. Introduction and definitions (pp. 720-721) | html | pdf |
    - 3.2.1.2. Crystallographic point groups (pp. 721-731) | html | pdf |
      - 3.2.1.2.1. Description of point groups (pp. 721-722) | html | pdf |
      - 3.2.1.2.2. Crystal and point forms (pp. 722-727) | html | pdf |
      - 3.2.1.2.3. Description of crystal and point forms (pp. 727-729) | html | pdf |
      - 3.2.1.2.4. Notes on crystal and point forms (pp. 729-730) | html | pdf |
      - 3.2.1.2.5. Names and symbols of the crystal classes (pp. 730-731) | html | pdf |
    - 3.2.1.3. Subgroups and supergroups of the crystallographic point groups (p. 731) | html | pdf |
    - 3.2.1.4. Noncrystallographic point groups (pp. 731-737) | html | pdf |
      - 3.2.1.4.1. Description of general point groups (pp. 731-733) | html | pdf |
      - 3.2.1.4.2. The two icosahedral groups (pp. 733-735) | html | pdf |
      - 3.2.1.4.3. Sub- and supergroups of the general point groups (pp. 735-737) | html | pdf |
  - 3.2.2. Point-group symmetry and physical properties of crystals (pp. 737-741) | html | pdf |
    H. Klapper and Th. Hahn
    - 3.2.2.1. General restrictions on physical properties imposed by symmetry (pp. 737-739) | html | pdf |
      - 3.2.2.1.1. Neumann's principle (p. 737) | html | pdf |
      - 3.2.2.1.2. Curie's principle (pp. 737-738) | html | pdf |
      - 3.2.2.1.3. Enantiomorphism, enantiomerism, chirality, dissymmetry (pp. 738-739) | html | pdf |
      - 3.2.2.1.4. Polar directions, polar axes, polar point groups (p. 739) | html | pdf |
    - 3.2.2.2. Morphology (pp. 739-740) | html | pdf |
    - 3.2.2.3. Etch figures (p. 740) | html | pdf |
    - 3.2.2.4. Optical properties (pp. 740-741) | html | pdf |
      - 3.2.2.4.1. Refraction (p. 740) | html | pdf |
      - 3.2.2.4.2. Optical activity (pp. 740-741) | html | pdf |
      - 3.2.2.4.3. Second-harmonic generation (SHG) (p. 741) | html | pdf |
    - 3.2.2.5. Pyroelectricity and ferroelectricity (p. 741) | html | pdf |
    - 3.2.2.6. Piezoelectricity (p. 741) | html | pdf |
  - 3.2.3. Tables of the crystallographic point-group types (pp. 742-771) | html | pdf |
    H. Klapper, Th. Hahn and M. I. Aroyo
  - 3.2.4. Molecular symmetry (pp. 772-776) | html | pdf |
    U. Müller
    - 3.2.4.1. Introduction (p. 772) | html | pdf |
    - 3.2.4.2. Definitions (pp. 772-773) | html | pdf |
    - 3.2.4.3. Tables of the point groups (pp. 773-774) | html | pdf |
    - 3.2.4.4. Polymeric molecules (pp. 774-775) | html | pdf |
    - 3.2.4.5. Enantiomorphism and chirality (pp. 775-776) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 3.2.1.1. The 47 crystal forms that crystals may take (p. 727) | html | pdf |
    - Fig. 3.2.1.2. Maximal subgroups and minimal supergroups of the two-dimensional crystallographic point groups (p. 731) | html | pdf |
    - Fig. 3.2.1.3. Maximal subgroups and minimal supergroups of the three-dimensional crystallographic point groups (p. 732) | html | pdf |
    - Fig. 3.2.1.4. Subgroups and supergroups of the two-dimensional general point groups (p. 735) | html | pdf |
    - Fig. 3.2.1.5. The subgroups of the two-dimensional general point groups 16mm (4N-gonal system) and 18mm [-gonal system, including the -gonal groups] (p. 735) | html | pdf |
    - Fig. 3.2.1.6. Subgroups and supergroups of the three-dimensional general point groups (p. 736) | html | pdf |
    - Fig. 3.2.4.1. P-helix of isotactic poly-4-methyl-1-pentene (form I) with a 7/2 helix which corresponds to a helix in Hermann–Mauguin notation (p. 774) | html | pdf |
  - Tables
    - Table 3.2.1.1. The ten two-dimensional crystallographic point groups, arranged according to crystal system (p. 720) | html | pdf |
    - Table 3.2.1.2. The 32 three-dimensional crystallographic point groups, arranged according to crystal system (cf. Chapter 2.1 ) (p. 721) | html | pdf |
    - Table 3.2.1.3. The 47 crystallographic face and point forms, their names, eigensymmetries, and their occurrence in the crystallographic point groups (generating point groups) (pp. 724-726) | html | pdf |
    - Table 3.2.1.4. Names and symbols of the 32 crystal classes (p. 728) | html | pdf |
    - Table 3.2.1.5. Classes of general point groups in two dimensions (N = integer $[\geq]$ 0) (p. 733) | html | pdf |
    - Table 3.2.1.6. Classes of general point groups in three dimensions (N = integer $[\geq]$ 0) (p. 734) | html | pdf |
    - Table 3.2.2.1. The 11 Laue classes, the 21 noncentrosymmetric crystallographic point groups (crystal classes) and the occurrence (+) of specific crystal properties (p. 738) | html | pdf |
    - Table 3.2.2.2. Polar axes and nonpolar directions in the 21 noncentrosymmetric crystal classes (p. 739) | html | pdf |
    - Table 3.2.2.3. Categories of crystal systems distinguished according to the different forms of the indicatrix (p. 740) | html | pdf |
    - Table 3.2.3.1. The ten two-dimensional crystallographic point groups (pp. 742-744) | html | pdf |
    - Table 3.2.3.2. The 32 three-dimensional crystallographic point groups (pp. 745-769) | html | pdf |
    - Table 3.2.3.3. The two icosahedral point groups (pp. 770-771) | html | pdf |
    - Table 3.2.4.1. Classes of rod groups (p. 775) | html | pdf |
- 3.3. Space-group symbols and their use (pp. 777-791) | html | pdf | chapter contents |
  H. Burzlaff and H. Zimmermann
  - 3.3.1. Point-group symbols (pp. 777-779) | html | pdf |
    - 3.3.1.1. Introduction (p. 777) | html | pdf |
    - 3.3.1.2. Schoenflies symbols (p. 777) | html | pdf |
    - 3.3.1.3. Shubnikov symbols (p. 777) | html | pdf |
    - 3.3.1.4. Hermann–Mauguin symbols (pp. 777-779) | html | pdf |
      - 3.3.1.4.1. Symmetry directions (p. 777) | html | pdf |
      - 3.3.1.4.2. Full Hermann–Mauguin symbols (p. 778) | html | pdf |
      - 3.3.1.4.3. Short symbols and generators (pp. 778-779) | html | pdf |
  - 3.3.2. Space-group symbols (pp. 779-780) | html | pdf |
    - 3.3.2.1. Introduction (p. 779) | html | pdf |
    - 3.3.2.2. Schoenflies symbols (p. 779) | html | pdf |
    - 3.3.2.3. The role of translation parts in the Shubnikov and Hermann–Mauguin symbols (p. 779) | html | pdf |
    - 3.3.2.4. Shubnikov symbols (pp. 779-780) | html | pdf |
    - 3.3.2.5. International short symbols (p. 780) | html | pdf |
  - 3.3.3. Properties of the international symbols (pp. 780-790) | html | pdf |
    - 3.3.3.1. Derivation of the space group from the short symbol (pp. 780-781) | html | pdf |
    - 3.3.3.2. Derivation of the full symbol from the short symbol (p. 781) | html | pdf |
    - 3.3.3.3. Non-symbolized symmetry elements (pp. 781-782) | html | pdf |
    - 3.3.3.4. Standardization rules for short symbols (p. 782) | html | pdf |
    - 3.3.3.5. Systematic absences (pp. 789-790) | html | pdf |
    - 3.3.3.6. Generalized symmetry (p. 790) | html | pdf |
  - 3.3.4. Changes introduced in space-group symbols since 1935 (p. 790) | html | pdf |
  - References | html | pdf |
  - Tables
    - Table 3.3.1.1. International (Hermann–Mauguin) and Shubnikov symbols for symmetry elements (p. 777) | html | pdf |
    - Table 3.3.1.2. Representatives for the sets of lattice symmetry directions in the various crystal families (p. 778) | html | pdf |
    - Table 3.3.1.3. Point-group symbols (p. 778) | html | pdf |
    - Table 3.3.2.1. Symbols of glide planes in the Shubnikov and Hermann–Mauguin space-group symbols (p. 779) | html | pdf |
    - Table 3.3.3.1. Standard space-group symbols (pp. 783-789) | html | pdf |
- 3.4. Lattice complexes (pp. 792-825) | html | pdf | chapter contents |
  W. Fischer and E. Koch
  - 3.4.1. The concept of lattice complexes and limiting complexes (pp. 792-796) | html | pdf |
    - 3.4.1.1. Introduction (p. 792) | html | pdf |
    - 3.4.1.2. Crystallographic orbits, Wyckoff positions, Wyckoff sets and types of Wyckoff set (pp. 792-793) | html | pdf |
    - 3.4.1.3. Point configurations and lattice complexes, reference symbols (pp. 793-794) | html | pdf |
    - 3.4.1.4. Limiting complexes and comprehensive complexes (pp. 794-795) | html | pdf |
    - 3.4.1.5. Additional properties of lattice complexes (pp. 795-796) | html | pdf |
      - 3.4.1.5.1. The degrees of freedom (p. 795) | html | pdf |
      - 3.4.1.5.2. Weissenberg complexes (pp. 795-796) | html | pdf |
  - 3.4.2. The concept of characteristic and non-characteristic orbits, comparison with the lattice-complex concept (pp. 796-798) | html | pdf |
    - 3.4.2.1. Definitions (p. 796) | html | pdf |
    - 3.4.2.2. Comparison of the concepts of lattice complexes and orbit types (pp. 796-798) | html | pdf |
  - 3.4.3. Descriptive lattice-complex symbols and the assignment of Wyckoff positions to lattice complexes (pp. 798-800) | html | pdf |
    - 3.4.3.1. Descriptive symbols (pp. 798-800) | html | pdf |
      - 3.4.3.1.1. Introduction (p. 798) | html | pdf |
      - 3.4.3.1.2. Invariant lattice complexes (pp. 798-799) | html | pdf |
      - 3.4.3.1.3. Lattice complexes with degrees of freedom (pp. 799-800) | html | pdf |
      - 3.4.3.1.4. Properties of the descriptive symbols (p. 800) | html | pdf |
    - 3.4.3.2. Assignment of Wyckoff positions to Wyckoff sets and to lattice complexes (p. 800) | html | pdf |
  - 3.4.4. Applications of the lattice-complex concept (pp. 800-824) | html | pdf |
    - 3.4.4.1. Geometrical properties of point configurations (pp. 800-823) | html | pdf |
    - 3.4.4.2. Relations between crystal structures (p. 823) | html | pdf |
    - 3.4.4.3. Reflection conditions (p. 823) | html | pdf |
    - 3.4.4.4. Phase transitions (pp. 823-824) | html | pdf |
    - 3.4.4.5. Incorrect space-group assignment (p. 824) | html | pdf |
    - 3.4.4.6. Application of descriptive lattice-complex symbols (p. 824) | html | pdf |
    - 3.4.4.7. Weissenberg complexes (p. 824) | html | pdf |
  - References | html | pdf |
  - Tables
    - Table 3.4.1.1. Reference symbols of the 31 Weissenberg complexes with f ≥ 1 degrees of freedom in $[{\bb E}^3]$ (p. 796) | html | pdf |
    - Table 3.4.2.1. Reference symbols of the 28 lattice complexes with f ≥ 1 degrees of freedom without any limiting complex (p. 797) | html | pdf |
    - Table 3.4.3.1. Descriptive symbols of invariant lattice complexes in their characteristic Wyckoff position (p. 798) | html | pdf |
    - Table 3.4.3.2. Plane groups: assignment of Wyckoff positions to Wyckoff sets and to lattice complexes (p. 801) | html | pdf |
    - Table 3.4.3.3. Space groups: assignment of Wyckoff positions to Wyckoff sets and to lattice complexes (pp. 802-822) | html | pdf |
- 3.5. Normalizers of space groups and their use in crystallography (pp. 826-851) | html | pdf | chapter contents |
  E. Koch, W. Fischer and U. Müller
  - 3.5.1. Introduction and definitions (pp. 826-827) | html | pdf |
    E. Koch, W. Fischer and U. Müller
    - 3.5.1.1. Introduction (p. 826) | html | pdf |
    - 3.5.1.2. Definitions (pp. 826-827) | html | pdf |
  - 3.5.2. Euclidean and affine normalizers of plane groups and space groups (pp. 827-838) | html | pdf |
    E. Koch, W. Fischer and U. Müller
    - 3.5.2.1. Euclidean normalizers of plane groups and space groups (pp. 827-830) | html | pdf |
    - 3.5.2.2. Affine normalizers of plane groups and space groups (pp. 830-838) | html | pdf |
  - 3.5.3. Examples of the use of normalizers (pp. 838-850) | html | pdf |
    E. Koch and W. Fischer
    - 3.5.3.1. Introduction (p. 838) | html | pdf |
    - 3.5.3.2. Equivalent point configurations, equivalent Wyckoff positions and equivalent descriptions of crystal structures (pp. 838-849) | html | pdf |
    - 3.5.3.3. Equivalent lists of structure factors (p. 849) | html | pdf |
    - 3.5.3.4. Euclidean- and affine-equivalent sub- and supergroups (pp. 849-850) | html | pdf |
    - 3.5.3.5. Reduction of the parameter regions to be considered for geometrical studies of point configurations (p. 850) | html | pdf |
  - 3.5.4. Normalizers of point groups (p. 851) | html | pdf |
    E. Koch and W. Fischer
  - References | html | pdf |
  - Figures
    - Fig. 3.5.2.1. Parameter range for space groups of types $[P2, P2_{1}, Pm, P2/m]$ and $[P2_{1}/m]$ (plane groups of types p1 and p2) (p. 827) | html | pdf |
    - Fig. 3.5.2.2. Parameter range for space groups of types C2, Pc, Cm, Cc, , , $[P2_{1}/c]$ and (p. 827) | html | pdf |
    - Fig. 3.5.2.3. Parameter range for space groups of types C2, Pc, Cm, Cc, , , $[P2_{1}/c]$ and (p. 828) | html | pdf |
    - Fig. 3.5.2.4. Parameter range for space groups of types C2, Pc, Cm, Cc, , , $[P2_{1}/c]$ and (p. 828) | html | pdf |
  - Tables
    - Table 3.5.2.1. Euclidean normalizers of the plane groups (p. 829) | html | pdf |
    - Table 3.5.2.2. Euclidean normalizers of the triclinic space groups (p. 830) | html | pdf |
    - Table 3.5.2.3. Euclidean and chirality-preserving Euclidean normalizers of the monoclinic space groups (pp. 831-838) | html | pdf |
    - Table 3.5.2.4. Euclidean and chirality-preserving Euclidean normalizers of the orthorhombic space groups (pp. 839-841) | html | pdf |
    - Table 3.5.2.5. Euclidean and chirality-preserving Euclidean normalizers of the tetragonal, trigonal, hexagonal and cubic space groups (pp. 842-846) | html | pdf |
    - Table 3.5.2.6. Affine normalizers of the triclinic and monoclinic space groups (p. 847) | html | pdf |
    - Table 3.5.2.7. Matrices and columns used in Table 3.5.2.6 for the description of the affine normalizers of monoclinic and triclinic space groups (p. 847) | html | pdf |
    - Table 3.5.3.1. Changes of structure-factor phases for the equivalent descriptions of a crystal structure in F222 (p. 849) | html | pdf |
    - Table 3.5.4.1. Normalizers of the two-dimensional point groups with respect to the full isometry group of the circle (p. 851) | html | pdf |
    - Table 3.5.4.2. Normalizers of the three-dimensional point groups with respect to the full isometry group of the sphere (p. 851) | html | pdf |
- 3.6. Magnetic subperiodic groups and magnetic space groups (pp. 852-865) | html | pdf | chapter contents |
  D. B. Litvin
  - 3.6.1. Introduction (p. 852) | html | pdf |
  - 3.6.2. Survey of magnetic subperiodic groups and magnetic space groups (pp. 852-857) | html | pdf |
    - 3.6.2.1. Reduced magnetic superfamilies of magnetic groups (pp. 852-853) | html | pdf |
    - 3.6.2.2. Survey of magnetic point groups, magnetic subperiodic groups and magnetic space groups (pp. 853-857) | html | pdf |
      - 3.6.2.2.1. Magnetic group type serial number (p. 853) | html | pdf |
      - 3.6.2.2.2. Magnetic group type symbol (pp. 853-854) | html | pdf |
      - 3.6.2.2.3. Standard set of coset representatives (p. 854) | html | pdf |
      - 3.6.2.2.4. Opechowski–Guccione magnetic group type symbols and the standard set of coset representatives (pp. 854-856) | html | pdf |
      - 3.6.2.2.5. Symbol of the subgroup $[{\cal D}]$ of index 2 of $[{\cal F}({\cal D})]$ (pp. 856-857) | html | pdf |
  - 3.6.3. Tables of properties of magnetic groups (pp. 857-863) | html | pdf |
    - 3.6.3.1. Lattice diagram (p. 857) | html | pdf |
    - 3.6.3.2. Heading (pp. 857-858) | html | pdf |
    - 3.6.3.3. Diagrams of symmetry elements and of the general positions (pp. 859-861) | html | pdf |
    - 3.6.3.4. Origin (p. 861) | html | pdf |
    - 3.6.3.5. Asymmetric unit (p. 861) | html | pdf |
    - 3.6.3.6. Symmetry operations (pp. 861-862) | html | pdf |
    - 3.6.3.7. Abbreviated headline (p. 862) | html | pdf |
    - 3.6.3.8. Generators selected (p. 862) | html | pdf |
    - 3.6.3.9. General and special positions with spins (magnetic moments) (p. 862) | html | pdf |
    - 3.6.3.10. Symmetry of special projections (pp. 862-863) | html | pdf |
  - 3.6.4. Comparison of OG and BNS magnetic group type symbols (p. 863) | html | pdf |
  - 3.6.5. Maximal subgroups of index ≤ 4 (pp. 863-864) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 3.6.3.1. Table of properties of the three-dimensional magnetic space group 51.13.399 P_2bm′ma (pp. 858-859) | html | pdf |
    - Fig. 3.6.3.2. Lattice diagrams of (a) the three-dimensional magnetic space group 26.1.168 $[{\cal F} = Pmc2_1]$ and (b) the three-dimensional magnetic space group 26.10.177 $[{\cal M}_R = P_{2b} m'c'2_1= {\cal F}({\cal D}) = Pmc2_1(Pca2_1)]$ (p. 858) | html | pdf |
    - Fig. 3.6.3.3. Symmetry diagram of P4₁2′2′ (p. 860) | html | pdf |
    - Fig. 3.6.3.4. General-position diagram of P4₁2′2′ (p. 860) | html | pdf |
    - Fig. 3.6.3.5. Diagrams of the magnetic space group P4₁221′ (p. 860) | html | pdf |
    - Fig. 3.6.3.6. General-position diagram of rod group 2.3.29 P2/c′11 (p. 860) | html | pdf |
  - Tables
    - Table 3.6.2.1. Numbers of types of groups in the reduced magnetic superfamilies of one-, two- and three-dimensional crystallographic point groups, subperiodic groups and space groups (p. 853) | html | pdf |
    - Table 3.6.2.2. Examples of the format of the survey of magnetic groups of three-dimensional magnetic space-group types (p. 853) | html | pdf |
    - Table 3.6.3.1. Symmetry operations of magnetic space group 51.14.400 P_2bmma′ (p. 861) | html | pdf |
    - Table 3.6.3.2. General positions of magnetic space group 51.14.400 P_2bmma′ (p. 862) | html | pdf |
    - Table 3.6.4.1. Comparisons of three-dimensional OG and BNS magnetic group type symbols (p. 863) | html | pdf |

International Tables for CrystallographyVolume A: Space-group symmetry

Edited by M. I. Aroyo

Contents

International Tables for Crystallography
Volume A: Space-group symmetry