(International Tables) Space-group symmetry

International Tables for Crystallography
Volume A: Space-group symmetry

First online edition (2006) ISBN: 978-0-7923-6590-7 doi: 10.1107/97809553602060000100

Edited by Th. Hahn

Foreword to the Fifth, Revised Edition  (p. xv) | html | pdf |
Th. Hahn
Preface  (pp. xvi-xvii) | html | pdf |
Th. Hahn
Computer Production of Volume A  (pp. xix-xx) | html | pdf |
M. I. Aroyo and P. B. Konstantinov

Symbols and terms used in this volume
- 1.1. Printed symbols for crystallographic items (pp. 2-3) | html | pdf | chapter contents |
  Th. Hahn
  - 1.1.1. Vectors, coefficients and coordinates (p. 2) | html | pdf |
  - 1.1.2. Directions and planes (p. 2) | html | pdf |
  - 1.1.3. Reciprocal space (p. 2) | html | pdf |
  - 1.1.4. Functions (p. 2) | html | pdf |
  - 1.1.5. Spaces (p. 3) | html | pdf |
  - 1.1.6. Motions and matrices (p. 3) | html | pdf |
  - 1.1.7. Groups (p. 3) | html | pdf |
- 1.2. Printed symbols for conventional centring types (p. 4) | html | pdf | chapter contents |
  Th. Hahn
  - 1.2.1. Printed symbols for the conventional centring types of one-, two- and three-dimensional cells (p. 4) | html | pdf |
  - 1.2.2. Notes on centred cells (p. 4) | html | pdf |
  - References | html | pdf |
- 1.3. Printed symbols for symmetry elements (pp. 5-6) | html | pdf | chapter contents |
  Th. Hahn
  - 1.3.1. Printed symbols for symmetry elements and for the corresponding symmetry operations in one, two and three dimensions (p. 5) | html | pdf |
  - 1.3.2. Notes on symmetry elements and symmetry operations (p. 6) | html | pdf |
  - References | html | pdf |
- 1.4. Graphical symbols for symmetry elements in one, two and three dimensions (pp. 7-11) | html | pdf | chapter contents |
  Th. Hahn
  - 1.4.1. Symmetry planes normal to the plane of projection (three dimensions) and symmetry lines in the plane of the figure (two dimensions) (p. 7) | html | pdf |
  - 1.4.2. Symmetry planes parallel to the plane of projection (p. 7) | html | pdf |
  - 1.4.3. Symmetry planes inclined to the plane of projection (in cubic space groups of classes $[\overline{4}{3m}]$ and $[m\overline{3}m]$ only) (p. 8) | html | pdf |
  - 1.4.4. Notes on graphical symbols of symmetry planes (p. 8) | html | pdf |
  - 1.4.5. Symmetry axes normal to the plane of projection and symmetry points in the plane of the figure (p. 9) | html | pdf |
  - 1.4.6. Symmetry axes parallel to the plane of projection (p. 10) | html | pdf |
  - 1.4.7. Symmetry axes inclined to the plane of projection (in cubic space groups only) (p. 10) | html | pdf |
  - References | html | pdf |

Guide to the use of the space-group tables
- 2.1. Classification and coordinate systems of space groups (pp. 14-16) | html | pdf | chapter contents |
  Th. Hahn and A. Looijenga-Vos
  - 2.1.1. Introduction (p. 14) | html | pdf |
  - 2.1.2. Space-group classification (p. 14) | html | pdf |
  - 2.1.3. Conventional coordinate systems and cells (pp. 14-16) | html | pdf |
  - References | html | pdf |
  - Tables
    - Table 2.1.2.1. Crystal families, crystal systems, conventional coordinate systems and Bravais lattices in one, two and three dimensions (p. 15) | html | pdf |
- 2.2. Contents and arrangement of the tables (pp. 17-41) | html | pdf | chapter contents |
  Th. Hahn and A. Looijenga-Vos
  - 2.2.1. General layout (p. 17) | html | pdf |
  - 2.2.2. Space groups with more than one description (p. 17) | html | pdf |
  - 2.2.3. Headline (pp. 17-18) | html | pdf |
  - 2.2.4. International (Hermann–Mauguin) symbols for plane groups and space groups (cf. Chapter 12.2 ) (pp. 18-19) | html | pdf |
    - 2.2.4.1. Present symbols (pp. 18-19) | html | pdf |
    - 2.2.4.2. Changes in Hermann–Mauguin space-group symbols as compared with the 1952 and 1935 editions of International Tables (p. 19) | html | pdf |
  - 2.2.5. Patterson symmetry (p. 19) | html | pdf |
  - 2.2.6. Space-group diagrams (pp. 20-24) | html | pdf |
    - 2.2.6.1. Plane groups (p. 20) | html | pdf |
    - 2.2.6.2. Triclinic space groups (p. 20) | html | pdf |
    - 2.2.6.3. Monoclinic space groups (cf. Sections 2.2.2 and 2.2.16) (p. 20) | html | pdf |
    - 2.2.6.4. Orthorhombic space groups and orthorhombic settings (pp. 20-23) | html | pdf |
    - 2.2.6.5. Tetragonal, trigonal P and hexagonal P space groups (p. 23) | html | pdf |
    - 2.2.6.6. Rhombohedral (trigonal R) space groups (p. 23) | html | pdf |
    - 2.2.6.7. Cubic space groups (p. 23) | html | pdf |
    - 2.2.6.8. Diagrams of the general position (pp. 23-24) | html | pdf |
  - 2.2.7. Origin (pp. 24-25) | html | pdf |
    - 2.2.7.1. Origin statement (pp. 24-25) | html | pdf |
  - 2.2.8. Asymmetric unit (pp. 25-26) | html | pdf |
  - 2.2.9. Symmetry operations (pp. 26-27) | html | pdf |
    - 2.2.9.1. Numbering scheme (p. 26) | html | pdf |
    - 2.2.9.2. Designation of symmetry operations (pp. 26-27) | html | pdf |
  - 2.2.10. Generators (p. 27) | html | pdf |
  - 2.2.11. Positions (pp. 27-28) | html | pdf |
  - 2.2.12. Oriented site-symmetry symbols (pp. 28-29) | html | pdf |
  - 2.2.13. Reflection conditions (pp. 29-32) | html | pdf |
    - 2.2.13.1. General reflection conditions (pp. 29-32) | html | pdf |
    - 2.2.13.2. Special or `extra' reflection conditions (p. 32) | html | pdf |
    - 2.2.13.3. Structural or non-space-group absences (p. 32) | html | pdf |
  - 2.2.14. Symmetry of special projections (pp. 33-35) | html | pdf |
    - 2.2.14.1. Data listed in the space-group tables (pp. 33-34) | html | pdf |
    - 2.2.14.2. Projections of centred cells (lattices) (p. 34) | html | pdf |
    - 2.2.14.3. Projections of symmetry elements (pp. 34-35) | html | pdf |
  - 2.2.15. Maximal subgroups and minimal supergroups (pp. 35-38) | html | pdf |
    - 2.2.15.1. Maximal non-isomorphic subgroups (pp. 35-36) | html | pdf |
    - 2.2.15.2. Maximal isomorphic subgroups of lowest index (cf. Part 13 ) (pp. 36-37) | html | pdf |
    - 2.2.15.3. Minimal non-isomorphic supergroups (p. 37) | html | pdf |
    - 2.2.15.4. Minimal isomorphic supergroups of lowest index (p. 37) | html | pdf |
    - 2.2.15.5. Note on basis vectors (pp. 37-38) | html | pdf |
  - 2.2.16. Monoclinic space groups (pp. 38-40) | html | pdf |
    - 2.2.16.1. Cell choices (p. 38) | html | pdf |
    - 2.2.16.2. Settings (pp. 38-39) | html | pdf |
    - 2.2.16.3. Cell choices and settings in the present tables (pp. 39-40) | html | pdf |
    - 2.2.16.4. Comparison with earlier editions of International Tables (p. 40) | html | pdf |
    - 2.2.16.5. Selection of monoclinic cell (p. 40) | html | pdf |
  - 2.2.17. Crystallographic groups in one dimension (p. 40) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 2.2.6.1. Triclinic space groups ( $[\hbox{\sf G}]$ = general-position diagram) (p. 21) | html | pdf |
    - Fig. 2.2.6.2. Monoclinic space groups, setting with unique axis b ( $[\hbox{\sf G}]$ = general-position diagram) (p. 21) | html | pdf |
    - Fig. 2.2.6.3. Monoclinic space groups, setting with unique axis c (p. 21) | html | pdf |
    - Fig. 2.2.6.4. Monoclinic space groups, cell choices 1, 2, 3 (p. 22) | html | pdf |
    - Fig. 2.2.6.5. Orthorhombic space groups (p. 22) | html | pdf |
    - Fig. 2.2.6.6. Orthorhombic space groups (p. 22) | html | pdf |
    - Fig. 2.2.6.7. Tetragonal space groups ( $[\hbox{\sf G}]$ = general-position diagram) (p. 23) | html | pdf |
    - Fig. 2.2.6.8. Trigonal P and hexagonal P space groups ( $[\hbox{\sf G}]$ = general-position diagram) (p. 23) | html | pdf |
    - Fig. 2.2.6.9. Rhombohedral R space groups (p. 23) | html | pdf |
    - Fig. 2.2.6.10. Cubic space groups ( $[\hbox{\sf G}]$ = general-position stereodiagrams) (p. 23) | html | pdf |
    - Fig. 2.2.8.1. Boundary planes of asymmetric units occurring in the space-group tables (p. 26) | html | pdf |
    - Fig. 2.2.16.1. The three primitive two-dimensional cells which are spanned by the shortest three translation vectors e, f, g in the monoclinic plane (p. 38) | html | pdf |
    - Fig. 2.2.17.1. The two line groups (one-dimensional space groups) (p. 40) | html | pdf |
  - Tables
    - Table 2.2.4.1. Lattice symmetry directions for two and three dimensions (p. 18) | html | pdf |
    - Table 2.2.4.2. Changes in Hermann–Mauguin symbols for two-dimensional groups (p. 19) | html | pdf |
    - Table 2.2.5.1. Patterson symmetries for two and three dimensions (p. 20) | html | pdf |
    - Table 2.2.6.1. Numbers of distinct projections and different Hermann–Mauguin symbols for the orthorhombic space groups (space-group number placed between parentheses), listed according to point group as indicated in the headline (p. 21) | html | pdf |
    - Table 2.2.7.1. Examples of origin statements (p. 25) | html | pdf |
    - Table 2.2.13.1. Integral reflection conditions for centred cells (lattices) (p. 29) | html | pdf |
    - Table 2.2.13.2. Zonal and serial reflection conditions for glide planes and screw axes (cf. Chapter 1.3 ) (pp. 30-31) | html | pdf |
    - Table 2.2.13.3. Reflection conditions for the plane groups (p. 31) | html | pdf |
    - Table 2.2.14.1. 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 Part 7 (p. 33) | html | pdf |
    - Table 2.2.14.2. Projections of crystallographic symmetry elements (p. 34) | html | pdf |
    - Table 2.2.16.1. Monoclinic setting symbols (unique axis is underlined) (p. 39) | html | pdf |
    - Table 2.2.16.2. Symbols for centring types and glide planes of monoclinic space groups (p. 39) | html | pdf |

Determination of space groups
- 3.1. Space-group determination and diffraction symbols (pp. 44-54) | html | pdf | chapter contents |
  A. Looijenga-Vos and M. J. Buerger
  - 3.1.1. Introduction (p. 44) | html | pdf |
  - 3.1.2. Laue class and cell (p. 44) | html | pdf |
  - 3.1.3. Reflection conditions and diffraction symbol (pp. 44-45) | html | pdf |
  - 3.1.4. Deduction of possible space groups (pp. 45-46) | html | pdf |
  - 3.1.5. Diffraction symbols and possible space groups (pp. 46-51) | html | pdf |
  - 3.1.6. Space-group determination by additional methods (pp. 51-53) | html | pdf |
    - 3.1.6.1. Chemical information (p. 51) | html | pdf |
    - 3.1.6.2. Point-group determination by methods other than the use of X-ray diffraction (p. 53) | html | pdf |
    - 3.1.6.3. Study of X-ray intensity distributions (p. 53) | html | pdf |
    - 3.1.6.4. Consideration of maxima in Patterson syntheses (p. 53) | html | pdf |
    - 3.1.6.5. Anomalous dispersion (p. 53) | html | pdf |
    - 3.1.6.6. Summary (p. 53) | html | pdf |
  - References | html | pdf |
  - Tables
    - Table 3.1.2.1. Laue classes and crystal systems (p. 44) | html | pdf |
    - Table 3.1.4.1. Reflection conditions, diffraction symbols and possible space groups (pp. 46-53) | html | pdf |

Synoptic tables of space-group symbols
- 4.1. Introduction to the synoptic tables (pp. 56-60) | html | pdf | chapter contents |
  E. F. Bertaut
  - 4.1.1. Introduction (p. 56) | html | pdf |
  - 4.1.2. Additional symmetry elements (pp. 56-60) | html | pdf |
    - 4.1.2.1. Integral translations (pp. 56-57) | html | pdf |
    - 4.1.2.2. Centring translations (pp. 57-59) | html | pdf |
    - 4.1.2.3. The priority rule (pp. 59-60) | html | pdf |
  - References | html | pdf |
  - Tables
    - Table 4.1.2.1. Location of additional symmetry element, if the translation vector t is perpendicular to the symmetry axis along or to the symmetry plane in (p. 57) | html | pdf |
    - Table 4.1.2.2. Additional symmetry elements and their locations, if the translation vector t is inclined to the symmetry axis or symmetry plane (p. 58) | html | pdf |
    - Table 4.1.2.3. Additional symmetry elements due to a centring vector t and their locations (p. 59) | html | pdf |
- 4.2. Symbols for plane groups (two-dimensional space groups) (p. 61) | html | pdf | chapter contents |
  E. F. Bertaut
  - 4.2.1. Arrangement of the tables (p. 61) | html | pdf |
  - 4.2.2. Additional symmetry elements and extended symbols (p. 61) | html | pdf |
  - 4.2.3. Multiple cells (p. 61) | html | pdf |
  - 4.2.4. Group–subgroup relations (p. 61) | html | pdf |
  - References | html | pdf |
  - Tables
    - Table 4.2.1.1. Index of symbols for plane groups (p. 61) | html | pdf |
- 4.3. Symbols for space groups (pp. 62-76) | html | pdf | chapter contents |
  E. F. Bertaut
  - 4.3.1. Triclinic system (p. 62) | html | pdf |
  - 4.3.2. Monoclinic system (pp. 62-68) | html | pdf |
    - 4.3.2.1. Historical note and arrangement of the tables (p. 62) | html | pdf |
    - 4.3.2.2. Transformation of space-group symbols (p. 62) | html | pdf |
    - 4.3.2.3. Group–subgroup relations (pp. 62-68) | html | pdf |
  - 4.3.3. Orthorhombic system (pp. 68-71) | html | pdf |
    - 4.3.3.1. Historical note and arrangement of the tables (pp. 68-70) | html | pdf |
    - 4.3.3.2. Group–subgroup relations (pp. 70-71) | html | pdf |
      - 4.3.3.2.1. Maximal non-isomorphic k subgroups of type IIa (decentred) (p. 70) | html | pdf |
      - 4.3.3.2.2. Maximal t subgroups of type I (p. 71) | html | pdf |
  - 4.3.4. Tetragonal system (pp. 71-73) | html | pdf |
    - 4.3.4.1. Historical note and arrangement of the tables (p. 71) | html | pdf |
    - 4.3.4.2. Relations between symmetry elements (p. 71) | html | pdf |
    - 4.3.4.3. Additional symmetry elements (p. 71) | html | pdf |
    - 4.3.4.4. Multiple cells (pp. 71-72) | html | pdf |
    - 4.3.4.5. Group–subgroup relations (pp. 72-73) | html | pdf |
      - 4.3.4.5.1. Maximal k subgroups (p. 72) | html | pdf |
      - 4.3.4.5.2. Maximal t subgroups (pp. 72-73) | html | pdf |
  - 4.3.5. Trigonal and hexagonal systems (pp. 73-75) | html | pdf |
    - 4.3.5.1. Historical note (p. 73) | html | pdf |
    - 4.3.5.2. Primitive cells (p. 73) | html | pdf |
    - 4.3.5.3. Multiple cells (pp. 73-74) | html | pdf |
    - 4.3.5.4. Relations between symmetry elements (p. 74) | html | pdf |
    - 4.3.5.5. Additional symmetry elements (p. 74) | html | pdf |
    - 4.3.5.6. Group–subgroup relations (pp. 74-75) | html | pdf |
      - 4.3.5.6.1. Maximal k subgroups (p. 74) | html | pdf |
      - 4.3.5.6.2. Maximal t subgroups (pp. 74-75) | html | pdf |
  - 4.3.6. Cubic system (pp. 75-76) | html | pdf |
    - 4.3.6.1. Historical note and arrangement of tables (p. 75) | html | pdf |
    - 4.3.6.2. Relations between symmetry elements (p. 75) | html | pdf |
    - 4.3.6.3. Additional symmetry elements (p. 75) | html | pdf |
    - 4.3.6.4. Group–subgroup relations (pp. 75-76) | html | pdf |
      - 4.3.6.4.1. Maximal k subgroups (p. 75) | html | pdf |
      - 4.3.6.4.2. Maximal t subgroups (pp. 75-76) | html | pdf |
  - References | html | pdf |
  - Tables
    - Table 4.3.2.1. Index of symbols for space groups for various settings and cells (pp. 63-69) | html | pdf |

Transformations in crystallography
- 5.1. Transformations of the coordinate system (unit-cell transformations) (pp. 78-85) | html | pdf | chapter contents |
  H. Arnold
  - 5.1.1. Introduction (p. 78) | html | pdf |
  - 5.1.2. Matrix notation (p. 78) | html | pdf |
  - 5.1.3. General transformation (pp. 78-85) | html | pdf |
  - Figures
    - Fig. 5.1.3.1. General affine transformation (p. 79) | html | pdf |
    - Fig. 5.1.3.2. Monoclinic centred lattice, projected along the unique axis (p. 83) | html | pdf |
    - Fig. 5.1.3.3. Body-centred cell I with a, b, c and a corresponding primitive cell P with a′, b′, c′ (p. 83) | html | pdf |
    - Fig. 5.1.3.4. Face-centred cell F with a, b, c and a corresponding primitive cell P with a′, b′, c′ (p. 83) | html | pdf |
    - Fig. 5.1.3.5. Tetragonal lattices, projected along $[[00\bar{1}]]$ (p. 84) | html | pdf |
    - Fig. 5.1.3.6. Unit cells in the rhombohedral lattice (p. 84) | html | pdf |
    - Fig. 5.1.3.7. Hexagonal lattice projected along $[[00\bar{1}]]$ (p. 84) | html | pdf |
    - Fig. 5.1.3.8. Hexagonal lattice projected along $[[00\bar{1}]]$ (p. 85) | html | pdf |
    - Fig. 5.1.3.9. Rhombohedral lattice with a triple hexagonal unit cell a, b, c in obverse setting (p. 85) | html | pdf |
    - Fig. 5.1.3.10. Rhombohedral lattice with primitive rhombohedral cell a, b, c and the three centred monoclinic cells (p. 85) | html | pdf |
  - Tables
    - Table 5.1.3.1. Selected 3 × 3 transformation matrices $[{\bi P}]$ and $[{\bi Q} = {\bi P}^{ -1}]$ (pp. 80-83) | html | pdf |
- 5.2. Transformations of symmetry operations (motions) (pp. 86-89) | html | pdf | chapter contents |
  H. Arnold
  - 5.2.1. Transformations (p. 86) | html | pdf |
  - 5.2.2. Invariants (pp. 86-87) | html | pdf |
    - 5.2.2.1. Position vector (p. 87) | html | pdf |
    - 5.2.2.2. Modulus of position vector (p. 87) | html | pdf |
    - 5.2.2.3. Metric matrix (p. 87) | html | pdf |
    - 5.2.2.4. Scalar product (p. 87) | html | pdf |
  - 5.2.3. Example: low cristobalite and high cristobalite (pp. 87-89) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 5.2.3.1. Positions of silicon atoms in the low-cristobalite structure, projected along $[[00\bar{1}]]$ (p. 88) | html | pdf |

The 17 plane groups (two-dimensional space groups)
- 6.1. The 17 plane groups (two-dimensional space groups) (pp. 92-109) | html | | chapter contents |
  - Plane group 1, p1 (p. 92) | html | pdf |
  - Plane group 2, p2 (p. 93) | html | pdf |
  - Plane group 3, pm (p. 94) | html | pdf |
  - Plane group 4, pg (p. 95) | html | pdf |
  - Plane group 5, cm (p. 96) | html | pdf |
  - Plane group 6, p2mm (p. 97) | html | pdf |
  - Plane group 7, p2mg (p. 98) | html | pdf |
  - Plane group 8, p2gg (p. 99) | html | pdf |
  - Plane group 9, c2mm (p. 100) | html | pdf |
  - Plane group 10, p4 (p. 101) | html | pdf |
  - Plane group 11, p4mm (p. 102) | html | pdf |
  - Plane group 12, p4gm (p. 103) | html | pdf |
  - Plane group 13, p3 (p. 104) | html | pdf |
  - Plane group 14, p3m1 (p. 105) | html | pdf |
  - Plane group 15, p31m (p. 106) | html | pdf |
  - Plane group 16, p6 (p. 107) | html | pdf |
  - Plane group 17, p6mm (pp. 108-109) | html | pdf |

The 230 space groups
- 7.1. The 230 space groups (pp. 112-717) | html | | chapter contents |
  - Space group 1, P1 (pp. 112-113) | html | pdf |
  - Space group 2, P-1 (pp. 114-115) | html | pdf |
  - Space group 3, P2 (pp. 116-119) | html | pdf |
    - Space group 3, unique axis b, P2 (pp. 116-117) | html | pdf |
    - Space group 3, unique axis c, P2 (pp. 118-119) | html | pdf |
  - Space group 4, P2₁ (pp. 120-123) | html | pdf |
    - Space group 4, unique axis b, P2₁ (pp. 120-121) | html | pdf |
    - Space group 4, unique axis c, P2₁ (pp. 122-123) | html | pdf |
  - Space group 5, C2 (pp. 124-131) | html | pdf |
    - Space group 5, unique axis b, C2 (pp. 124-127) | html | pdf |
    - Space group 5, unique axis c, C2 (pp. 128-131) | html | pdf |
  - Space group 6, Pm (pp. 132-135) | html | pdf |
    - Space group 6, unique axis b, Pm (pp. 132-133) | html | pdf |
    - Space group 6, unique axis c, Pm (pp. 134-135) | html | pdf |
  - Space group 7, Pc (pp. 136-143) | html | pdf |
    - Space group 7, unique axis b, Pc (pp. 136-139) | html | pdf |
    - Space group 7, unique axis c, Pc (pp. 140-143) | html | pdf |
  - Space group 8, Cm (pp. 144-151) | html | pdf |
    - Space group 8, unique axis b, Cm (pp. 144-147) | html | pdf |
    - Space group 8, unique axis c, Cm (pp. 148-151) | html | pdf |
  - Space group 9, Cc (pp. 152-159) | html | pdf |
    - Space group 9, unique axis b, Cc (pp. 152-155) | html | pdf |
    - Space group 9, unique axis c, Cc (pp. 156-159) | html | pdf |
  - Space group 10, P2/m (pp. 160-163) | html | pdf |
    - Space group 10, unique axis b, P2/m (pp. 160-161) | html | pdf |
    - Space group 10, unique axis c, P2/m (pp. 162-163) | html | pdf |
  - Space group 11, P2₁/m (pp. 164-167) | html | pdf |
    - Space group 11, unique axis b, P2₁/m (pp. 164-165) | html | pdf |
    - Space group 11, unique axis c, P2₁/m (pp. 166-167) | html | pdf |
  - Space group 12, C2/m (pp. 168-175) | html | pdf |
    - Space group 12, unique axis b, C2/m (pp. 168-171) | html | pdf |
    - Space group 12, unique axis c, C2/m (pp. 172-175) | html | pdf |
  - Space group 13, P2/c (pp. 176-183) | html | pdf |
    - Space group 13, unique axis b, P2/c (pp. 176-179) | html | pdf |
    - Space group 13, unique axis c, P2/c (pp. 180-183) | html | pdf |
  - Space group 14, P2₁/c (pp. 184-191) | html | pdf |
    - Space group 14, unique axis b, P2₁/c (pp. 184-187) | html | pdf |
    - Space group 14, unique axis c, P2₁/c (pp. 188-191) | html | pdf |
  - Space group 15, C2/c (pp. 192-199) | html | pdf |
    - Space group 15, unique axis b, C2/c (pp. 192-195) | html | pdf |
    - Space group 15, unique axis c, C2/c (pp. 196-199) | html | pdf |
  - Space group 16, P222 (pp. 200-201) | html | pdf |
  - Space group 17, P222₁ (pp. 202-203) | html | pdf |
  - Space group 18, P2₁2₁2 (pp. 204-205) | html | pdf |
  - Space group 19, P2₁2₁2₁ (pp. 206-207) | html | pdf |
  - Space group 20, C222₁ (pp. 208-209) | html | pdf |
  - Space group 21, C222 (pp. 210-211) | html | pdf |
  - Space group 22, F222 (pp. 212-213) | html | pdf |
  - Space group 23, I222 (pp. 214-215) | html | pdf |
  - Space group 24, I2₁2₁2₁ (pp. 216-217) | html | pdf |
  - Space group 25, Pmm2 (pp. 218-219) | html | pdf |
  - Space group 26, Pmc2₁ (pp. 220-221) | html | pdf |
  - Space group 27, Pcc2 (pp. 222-223) | html | pdf |
  - Space group 28, Pma2 (pp. 224-225) | html | pdf |
  - Space group 29, Pca2₁ (pp. 226-227) | html | pdf |
  - Space group 30, Pnc2 (pp. 228-229) | html | pdf |
  - Space group 31, Pmn2₁ (pp. 230-231) | html | pdf |
  - Space group 32, Pba2 (pp. 232-233) | html | pdf |
  - Space group 33, Pna2₁ (pp. 234-235) | html | pdf |
  - Space group 34, Pnn2 (pp. 236-237) | html | pdf |
  - Space group 35, Cmm2 (pp. 238-239) | html | pdf |
  - Space group 36, Cmc2₁ (pp. 240-241) | html | pdf |
  - Space group 37, Ccc2 (pp. 242-243) | html | pdf |
  - Space group 38, Amm2 (pp. 244-245) | html | pdf |
  - Space group 39, Aem2 (pp. 246-247) | html | pdf |
  - Space group 40, Ama2 (pp. 248-249) | html | pdf |
  - Space group 41, Aea2 (pp. 250-251) | html | pdf |
  - Space group 42, Fmm2 (pp. 252-253) | html | pdf |
  - Space group 43, Fdd2 (pp. 254-255) | html | pdf |
  - Space group 44, Imm2 (pp. 256-257) | html | pdf |
  - Space group 45, Iba2 (pp. 258-259) | html | pdf |
  - Space group 46, Ima2 (pp. 260-261) | html | pdf |
  - Space group 47, Pmmm (pp. 262-263) | html | pdf |
  - Space group 48, Pnnn (pp. 264-267) | html | pdf |
    - Space group 48, origin choice 1, Pnnn (pp. 264-265) | html | pdf |
    - Space group 48, origin choice 2, Pnnn (pp. 266-267) | html | pdf |
  - Space group 49, Pccm (pp. 268-269) | html | pdf |
  - Space group 50, Pban (pp. 270-273) | html | pdf |
    - Space group 50, origin choice 1, Pban (pp. 270-271) | html | pdf |
    - Space group 50, origin choice 2, Pban (pp. 272-273) | html | pdf |
  - Space group 51, Pmma (pp. 274-275) | html | pdf |
  - Space group 52, Pnna (pp. 276-277) | html | pdf |
  - Space group 53, Pmna (pp. 278-279) | html | pdf |
  - Space group 54, Pcca (pp. 280-281) | html | pdf |
  - Space group 55, Pbam (pp. 282-283) | html | pdf |
  - Space group 56, Pccn (pp. 284-285) | html | pdf |
  - Space group 57, Pbcm (pp. 286-287) | html | pdf |
  - Space group 58, Pnnm (pp. 288-289) | html | pdf |
  - Space group 59, Pmmn (pp. 290-293) | html | pdf |
    - Space group 59, origin choice 1, Pmmn (pp. 290-291) | html | pdf |
    - Space group 59, origin choice 2, Pmmn (pp. 292-293) | html | pdf |
  - Space group 60, Pbcn (pp. 294-295) | html | pdf |
  - Space group 61, Pbca (pp. 296-297) | html | pdf |
  - Space group 62, Pnma (pp. 298-299) | html | pdf |
  - Space group 63, Cmcm (pp. 300-301) | html | pdf |
  - Space group 64, Cmce (pp. 302-303) | html | pdf |
  - Space group 65, Cmmm (pp. 304-306) | html | pdf |
  - Space group 66, Cccm (pp. 308-309) | html | pdf |
  - Space group 67, Cmme (pp. 310-311) | html | pdf |
  - Space group 68, Ccce (pp. 312-315) | html | pdf |
    - Space group 68, origin choice 1, Ccce (pp. 312-313) | html | pdf |
    - Space group 68, origin choice 2, Ccce (pp. 314-315) | html | pdf |
  - Space group 69, Fmmm (pp. 316-318) | html | pdf |
  - Space group 70, Fddd (pp. 320-323) | html | pdf |
    - Space group 70, origin choice 1, Fddd (pp. 320-321) | html | pdf |
    - Space group 70, origin choice 2, Fddd (pp. 322-323) | html | pdf |
  - Space group 71, Immm (pp. 324-325) | html | pdf |
  - Space group 72, Ibam (pp. 326-327) | html | pdf |
  - Space group 73, Ibca (pp. 328-329) | html | pdf |
  - Space group 74, Imma (pp. 330-331) | html | pdf |
  - Space group 75, P4 (p. 332) | html | pdf |
  - Space group 76, P4₁ (p. 333) | html | pdf |
  - Space group 77, P4₂ (p. 334) | html | pdf |
  - Space group 78, P4₃ (p. 335) | html | pdf |
  - Space group 79, I4 (pp. 336-337) | html | pdf |
  - Space group 80, I4₁ (pp. 338-339) | html | pdf |
  - Space group 81, P-4 (pp. 340-341) | html | pdf |
  - Space group 82, I-4 (pp. 342-343) | html | pdf |
  - Space group 83, P4/m (pp. 344-345) | html | pdf |
  - Space group 84, P4₂/m (pp. 346-347) | html | pdf |
  - Space group 85, P4/n (pp. 348-351) | html | pdf |
    - Space group 85, origin choice 1, P4/n (pp. 348-349) | html | pdf |
    - Space group 85, origin choice 2, P4/n (pp. 350-351) | html | pdf |
  - Space group 86, P4₂/n (pp. 352-355) | html | pdf |
    - Space group 86, origin choice 1, P4₂/n (pp. 352-353) | html | pdf |
    - Space group 86, origin choice 2, P4₂/n (pp. 354-355) | html | pdf |
  - Space group 87, I4/m (pp. 356-357) | html | pdf |
  - Space group 88, I4₁/a (pp. 358-361) | html | pdf |
    - Space group 88, origin choice 1, I4₁/a (pp. 358-359) | html | pdf |
    - Space group 88, origin choice 2, I4₁/a (pp. 360-361) | html | pdf |
  - Space group 89, P422 (pp. 362-363) | html | pdf |
  - Space group 90, P42₁2 (pp. 364-365) | html | pdf |
  - Space group 91, P4₁22 (pp. 366-367) | html | pdf |
  - Space group 92, P4₁2₁2 (pp. 368-369) | html | pdf |
  - Space group 93, P4₂22 (pp. 370-371) | html | pdf |
  - Space group 94, P4₂2₁2 (pp. 372-373) | html | pdf |
  - Space group 95, P4₃22 (pp. 374-375) | html | pdf |
  - Space group 96, P4₃2₁2 (pp. 376-377) | html | pdf |
  - Space group 97, I422 (pp. 378-379) | html | pdf |
  - Space group 98, I4₁22 (pp. 380-381) | html | pdf |
  - Space group 99, P4mm (pp. 382-383) | html | pdf |
  - Space group 100, P4bm (pp. 384-385) | html | pdf |
  - Space group 101, P4₂cm (pp. 386-387) | html | pdf |
  - Space group 102, P4₂nm (pp. 388-389) | html | pdf |
  - Space group 103, P4cc (pp. 390-391) | html | pdf |
  - Space group 104, P4nc (pp. 392-393) | html | pdf |
  - Space group 105, P4₂mc (pp. 394-395) | html | pdf |
  - Space group 106, P4₂bc (pp. 396-397) | html | pdf |
  - Space group 107, I4mm (pp. 398-399) | html | pdf |
  - Space group 108, I4cm (pp. 400-401) | html | pdf |
  - Space group 109, I4₁md (pp. 402-403) | html | pdf |
  - Space group 110, I4₁cd (pp. 404-405) | html | pdf |
  - Space group 111, P-42m (pp. 406-407) | html | pdf |
  - Space group 112, P-42c (pp. 408-409) | html | pdf |
  - Space group 113, P-42₁m (pp. 410-411) | html | pdf |
  - Space group 114, P-42₁c (pp. 412-413) | html | pdf |
  - Space group 115, P-4m2 (pp. 414-415) | html | pdf |
  - Space group 116, P-4c2 (pp. 416-417) | html | pdf |
  - Space group 117, P-4b2 (pp. 418-419) | html | pdf |
  - Space group 118, P-4n2 (pp. 420-421) | html | pdf |
  - Space group 119, I-4m2 (pp. 422-423) | html | pdf |
  - Space group 120, I-4c2 (pp. 424-425) | html | pdf |
  - Space group 121, I-42m (pp. 426-427) | html | pdf |
  - Space group 122, I-42d (pp. 428-429) | html | pdf |
  - Space group 123, P4/mmm (pp. 430-431) | html | pdf |
  - Space group 124, P4/mcc (pp. 432-433) | html | pdf |
  - Space group 125, P4/nbm (pp. 434-437) | html | pdf |
    - Space group 125, origin choice 1, P4/nbm (pp. 434-435) | html | pdf |
    - Space group 125, origin choice 2, P4/nbm (pp. 436-437) | html | pdf |
  - Space group 126, P4/nnc (pp. 438-441) | html | pdf |
    - Space group 126, origin choice 1, P4/nnc (pp. 438-439) | html | pdf |
    - Space group 126, origin choice 2, P4/nnc (pp. 440-441) | html | pdf |
  - Space group 127, P4/mbm (pp. 442-443) | html | pdf |
  - Space group 128, P4/mnc (pp. 444-445) | html | pdf |
  - Space group 129, P4/nmm (pp. 446-449) | html | pdf |
    - Space group 129, origin choice 1, P4/nmm (pp. 446-447) | html | pdf |
    - Space group 129, origin choice 2, P4/nmm (pp. 448-449) | html | pdf |
  - Space group 130, P4/ncc (pp. 450-453) | html | pdf |
    - Space group 130, origin choice 1, P4/ncc (pp. 450-451) | html | pdf |
    - Space group 130, origin choice 2, P4/ncc (pp. 452-453) | html | pdf |
  - Space group 131, P4₂/mmc (pp. 454-455) | html | pdf |
  - Space group 132, P4₂/mcm (pp. 456-457) | html | pdf |
  - Space group 133, P4₂/nbc (pp. 458-461) | html | pdf |
    - Space group 133, origin choice 1, P4₂/nbc (pp. 458-459) | html | pdf |
    - Space group 133, origin choice 2, P4₂/nbc (pp. 460-461) | html | pdf |
  - Space group 134, P4₂/nnm (pp. 462-465) | html | pdf |
    - Space group 134, origin choice 1, P4₂/nnm (pp. 462-463) | html | pdf |
    - Space group 134, origin choice 2, P4₂/nnm (pp. 464-465) | html | pdf |
  - Space group 135, P4₂/mbc (pp. 466-467) | html | pdf |
  - Space group 136, P4₂/mnm (pp. 468-469) | html | pdf |
  - Space group 137, P4₂/nmc (pp. 470-473) | html | pdf |
    - Space group 137, origin choice 1, P4₂/nmc (pp. 470-471) | html | pdf |
    - Space group 137, origin choice 2, P4₂/nmc (pp. 472-473) | html | pdf |
  - Space group 138, P4₂/ncm (pp. 474-477) | html | pdf |
    - Space group 138, origin choice 1, P4₂/ncm (pp. 474-475) | html | pdf |
    - Space group 138, origin choice 2, P4₂/ncm (pp. 476-477) | html | pdf |
  - Space group 139, I4/mmm (pp. 478-479) | html | pdf |
  - Space group 140, I4/mcm (pp. 480-481) | html | pdf |
  - Space group 141, I4₁/amd (pp. 482-485) | html | pdf |
    - Space group 141, origin choice 1, I4₁/amd (pp. 482-483) | html | pdf |
    - Space group 141, origin choice 2, I4₁/amd (pp. 484-485) | html | pdf |
  - Space group 142, I4₁/acd (pp. 486-489) | html | pdf |
    - Space group 142, origin choice 1, I4₁/acd (pp. 486-487) | html | pdf |
    - Space group 142, origin choice 2, I4₁/acd (pp. 488-489) | html | pdf |
  - Space group 143, P3 (pp. 490-491) | html | pdf |
  - Space group 144, P3₁ (p. 492) | html | pdf |
  - Space group 145, P3₂ (p. 493) | html | pdf |
  - Space group 146, R3 (pp. 494-497) | html | pdf |
    - Space group 146, hexagonal axes, R3 (pp. 494-495) | html | pdf |
    - Space group 146, rhombohedral axes, R3 (pp. 496-497) | html | pdf |
  - Space group 147, P-3 (pp. 498-499) | html | pdf |
  - Space group 148, R-3 (pp. 500-503) | html | pdf |
    - Space group 148, hexagonal axes, R-3 (pp. 500-501) | html | pdf |
    - Space group 148, rhombohedral axes, R-3 (pp. 502-503) | html | pdf |
  - Space group 149, P312 (pp. 504-505) | html | pdf |
  - Space group 150, P321 (pp. 506-507) | html | pdf |
  - Space group 151, P3₁12 (pp. 508-509) | html | pdf |
  - Space group 152, P3₁21 (pp. 510-511) | html | pdf |
  - Space group 153, P3₂12 (pp. 512-513) | html | pdf |
  - Space group 154, P3₂21 (pp. 514-515) | html | pdf |
  - Space group 155, R32 (pp. 516-519) | html | pdf |
    - Space group 155, hexagonal axes, R32 (pp. 516-517) | html | pdf |
    - Space group 155, rhombohedral axes, R32 (pp. 518-519) | html | pdf |
  - Space group 156, P3m1 (pp. 520-521) | html | pdf |
  - Space group 157, P31m (pp. 522-523) | html | pdf |
  - Space group 158, P3c1 (pp. 524-525) | html | pdf |
  - Space group 159, P31c (pp. 526-527) | html | pdf |
  - Space group 160, R3m (pp. 528-531) | html | pdf |
    - Space group 160, hexagonal axes, R3m (pp. 528-529) | html | pdf |
    - Space group 160, rhombohedral axes, R3m (pp. 530-531) | html | pdf |
  - Space group 161, R3c (pp. 532-535) | html | pdf |
    - Space group 161, hexagonal axes, R3c (pp. 532-533) | html | pdf |
    - Space group 161, rhombohedral axes, R3c (pp. 534-535) | html | pdf |
  - Space group 162, P-31m (pp. 536-537) | html | pdf |
  - Space group 163, P-31c (pp. 538-539) | html | pdf |
  - Space group 164, P-3m1 (pp. 540-541) | html | pdf |
  - Space group 165, P-3c1 (pp. 542-543) | html | pdf |
  - Space group 166, R-3m (pp. 544-547) | html | pdf |
    - Space group 166, hexagonal axes, R-3m (pp. 544-546) | html | pdf |
    - Space group 166, rhombohedral axes, R-3m (pp. 546-547) | html | pdf |
  - Space group 167, R-3c (pp. 548-551) | html | pdf |
    - Space group 167, hexagonal axes, R-3c (pp. 548-550) | html | pdf |
    - Space group 167, rhombohedral axes, R-3c (pp. 550-551) | html | pdf |
  - Space group 168, P6 (pp. 552-553) | html | pdf |
  - Space group 169, P6₁ (p. 554) | html | pdf |
  - Space group 170, P6₅ (p. 555) | html | pdf |
  - Space group 171, P6₂ (p. 556) | html | pdf |
  - Space group 172, P6₄ (p. 557) | html | pdf |
  - Space group 173, P6₃ (pp. 558-559) | html | pdf |
  - Space group 174, P-6 (pp. 560-561) | html | pdf |
  - Space group 175, P6/m (pp. 562-563) | html | pdf |
  - Space group 176, P6₃/m (pp. 564-565) | html | pdf |
  - Space group 177, P622 (pp. 566-567) | html | pdf |
  - Space group 178, P6₁22 (pp. 568-569) | html | pdf |
  - Space group 179, P6₅22 (pp. 570-571) | html | pdf |
  - Space group 180, P6₂22 (pp. 572-573) | html | pdf |
  - Space group 181, P6₄22 (pp. 574-575) | html | pdf |
  - Space group 182, P6₃22 (pp. 576-577) | html | pdf |
  - Space group 183, P6mm (pp. 578-579) | html | pdf |
  - Space group 184, P6cc (pp. 580-581) | html | pdf |
  - Space group 185, P6₃cm (pp. 582-583) | html | pdf |
  - Space group 186, P6₃mc (pp. 584-585) | html | pdf |
  - Space group 187, P-6m2 (pp. 586-587) | html | pdf |
  - Space group 188, P-6c2 (pp. 588-589) | html | pdf |
  - Space group 189, P-62m (pp. 590-591) | html | pdf |
  - Space group 190, P-62c (pp. 592-593) | html | pdf |
  - Space group 191, P6/mmm (pp. 594-595) | html | pdf |
  - Space group 192, P6/mcc (pp. 596-597) | html | pdf |
  - Space group 193, P6₃/mcm (pp. 598-599) | html | pdf |
  - Space group 194, P6₃/mmc (pp. 600-601) | html | pdf |
  - Space group 195, P23 (pp. 602-603) | html | pdf |
  - Space group 196, F23 (pp. 604-606) | html | pdf |
  - Space group 197, I23 (pp. 608-609) | html | pdf |
  - Space group 198, P2₁3 (pp. 610-611) | html | pdf |
  - Space group 199, I2₁3 (pp. 612-613) | html | pdf |
  - Space group 200, Pm-3 (pp. 614-615) | html | pdf |
  - Space group 201, Pn-3 (pp. 616-619) | html | pdf |
    - Space group 201, origin choice 1, Pn-3 (pp. 616-617) | html | pdf |
    - Space group 201, origin choice 2, Pn-3 (pp. 618-619) | html | pdf |
  - Space group 202, Fm-3 (pp. 620-622) | html | pdf |
  - Space group 203, Fd-3 (pp. 623-627) | html | pdf |
    - Space group 203, origin choice 1, Fd-3 (pp. 623-625) | html | pdf |
    - Space group 203, origin choice 2, Fd-3 (pp. 626-627) | html | pdf |
  - Space group 204, Im-3 (pp. 628-629) | html | pdf |
  - Space group 205, Pa-3 (pp. 630-631) | html | pdf |
  - Space group 206, Ia-3 (pp. 632-633) | html | pdf |
  - Space group 207, P432 (pp. 634-635) | html | pdf |
  - Space group 208, P4₂32 (pp. 636-638) | html | pdf |
  - Space group 209, F432 (pp. 639-641) | html | pdf |
  - Space group 210, F4₁32 (pp. 642-644) | html | pdf |
  - Space group 211, I432 (pp. 645-647) | html | pdf |
  - Space group 212, P4₃32 (pp. 648-649) | html | pdf |
  - Space group 213, P4₁32 (pp. 650-651) | html | pdf |
  - Space group 214, I4₁32 (pp. 652-654) | html | pdf |
  - Space group 215, P-43m (pp. 656-657) | html | pdf |
  - Space group 216, F-43m (pp. 658-660) | html | pdf |
  - Space group 217, I-43m (pp. 661-663) | html | pdf |
  - Space group 218, P-43n (pp. 664-665) | html | pdf |
  - Space group 219, F-43c (pp. 666-668) | html | pdf |
  - Space group 220, I-43d (pp. 669-671) | html | pdf |
  - Space group 221, Pm-3m (pp. 672-674) | html | pdf |
  - Space group 222, Pn-3n (pp. 675-679) | html | pdf |
    - Space group 222, origin choice 1, Pn-3n (pp. 675-677) | html | pdf |
    - Space group 222, origin choice 2, Pn-3n (pp. 678-679) | html | pdf |
  - Space group 223, Pm-3n (pp. 680-682) | html | pdf |
  - Space group 224, Pn-3m (pp. 683-687) | html | pdf |
    - Space group 224, origin choice 1, Pn-3m (pp. 683-685) | html | pdf |
    - Space group 224, origin choice 2, Pn-3m (pp. 686-687) | html | pdf |
  - Space group 225, Fm-3m (pp. 688-691) | html | pdf |
  - Space group 226, Fm-3c (pp. 692-695) | html | pdf |
  - Space group 227, Fd-3m (pp. 696-703) | html | pdf |
    - Space group 227, origin choice 1, Fd-3m (pp. 696-699) | html | pdf |
    - Space group 227, origin choice 2, Fd-3m (pp. 700-703) | html | pdf |
  - Space group 228, Fd-3c (pp. 704-711) | html | pdf |
    - Space group 228, origin choice 1, Fd-3c (pp. 704-707) | html | pdf |
    - Space group 228, origin choice 2, Fd-3c (pp. 708-711) | html | pdf |
  - Space group 229, Im-3m (pp. 712-714) | html | pdf |
  - Space group 230, Ia-3d (pp. 715-717) | html | pdf |

Introduction to space-group symmetry
- 8.1. Basic concepts (pp. 720-725) | html | pdf | chapter contents |
  H. Wondratschek
  - 8.1.1. Introduction (p. 720) | html | pdf |
  - 8.1.2. Spaces and motions (pp. 720-722) | html | pdf |
  - 8.1.3. Symmetry operations and symmetry groups (p. 722) | html | pdf |
  - 8.1.4. Crystal patterns, vector lattices and point lattices (pp. 722-723) | html | pdf |
  - 8.1.5. Crystallographic symmetry operations (pp. 723-724) | html | pdf |
  - 8.1.6. Space groups and point groups (pp. 724-725) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 8.1.2.1. Representation of the point X with respect to origin O by the vector $[\overrightarrow{OX} = {\bf x}]$ (p. 721) | html | pdf |
    - Fig. 8.1.2.2. Relations between the different kinds of motions in E³ (p. 722) | html | pdf |
  - Tables
    - Table 8.1.1.1. Number of crystallographic classes for dimensions 1 to 6 (p. 720) | html | pdf |
- 8.2. Classifications of space groups, point groups and lattices (pp. 726-731) | html | pdf | chapter contents |
  H. Wondratschek
  - 8.2.1. Introduction (p. 726) | html | pdf |
  - 8.2.2. Space-group types (pp. 726-727) | html | pdf |
  - 8.2.3. Arithmetic crystal classes (pp. 727-728) | html | pdf |
  - 8.2.4. Geometric crystal classes (p. 728) | html | pdf |
  - 8.2.5. Bravais classes of matrices and Bravais types of lattices (lattice types) (pp. 728-729) | html | pdf |
  - 8.2.6. Bravais flocks of space groups (p. 729) | html | pdf |
  - 8.2.7. Crystal families (pp. 729-730) | html | pdf |
  - 8.2.8. Crystal systems and lattice systems (pp. 730-731) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 8.2.1.1. Classifications of space groups (p. 726) | html | pdf |
  - Tables
    - Table 8.2.8.1. Distribution of trigonal and hexagonal space groups into crystal systems and lattice systems (p. 730) | html | pdf |
- 8.3. Special topics on space groups (pp. 732-740) | html | pdf | chapter contents |
  H. Wondratschek
  - 8.3.1. Coordinate systems in crystallography (p. 732) | html | pdf |
  - 8.3.2. (Wyckoff) positions, site symmetries and crystallographic orbits (pp. 732-734) | html | pdf |
  - 8.3.3. Subgroups and supergroups of space groups (pp. 734-736) | html | pdf |
    - 8.3.3.1. Translationengleiche or t subgroups of a space group (p. 735) | html | pdf |
    - 8.3.3.2. Klassengleiche or k subgroups of a space group (p. 735) | html | pdf |
    - 8.3.3.3. Supergroups (pp. 735-736) | html | pdf |
  - 8.3.4. Sequence of space-group types (p. 736) | html | pdf |
  - 8.3.5. Space-group generators (pp. 736-738) | html | pdf |
    - 8.3.5.1. Selected order for non-translational generators (pp. 737-738) | html | pdf |
  - 8.3.6. Normalizers of space groups (pp. 738-739) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 8.3.3.1. Space group $[P6_{3}/mcm]$ with t subgroups of index [2] and [4] (p. 735) | html | pdf |
    - Fig. 8.3.6.1. Square with `symmetry elements' $[m_{x}]$ , $[m_{y}]$ , $[m_{d}]$ , $[m_{d'}]$ (mirror lines) and ♦ (fourfold rotation point) (p. 739) | html | pdf |
  - Tables
    - Table 8.3.4.1. Listing of space-group types according to their geometric and arithmetic crystal classes (p. 736) | html | pdf |
    - Table 8.3.5.1. Sequence of generators for the crystal classes (p. 737) | html | pdf |
    - Table 8.3.5.2. Example of a space-group generation $[{\cal G}: {P6_{1}22 \equiv D_{6}^{2}}]$ (No. 178) (p. 738) | html | pdf |

Crystal lattices
- 9.1. Bases, lattices, Bravais lattices and other classifications (pp. 742-749) | html | pdf | chapter contents |
  H. Burzlaff and H. Zimmermann
  - 9.1.1. Description and transformation of bases (p. 742) | html | pdf |
  - 9.1.2. Lattices (p. 742) | html | pdf |
  - 9.1.3. Topological properties of lattices (p. 742) | html | pdf |
  - 9.1.4. Special bases for lattices (pp. 742-743) | html | pdf |
  - 9.1.5. Remarks (p. 743) | html | pdf |
  - 9.1.6. Classifications (pp. 743-745) | html | pdf |
  - 9.1.7. Description of Bravais lattices (p. 745) | html | pdf |
  - 9.1.8. Delaunay reduction (pp. 745-749) | html | pdf |
  - 9.1.9. Example (p. 749) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 9.1.7.1. Conventional cells of the three-dimensional Bravais lattices (p. 744) | html | pdf |
  - Tables
    - Table 9.1.4.1. Lattice point-group symmetries (p. 742) | html | pdf |
    - Table 9.1.6.1. Representations of the five types of Voronoi polyhedra (p. 744) | html | pdf |
    - Table 9.1.7.1. Two-dimensional Bravais lattices (p. 745) | html | pdf |
    - Table 9.1.7.2. Three-dimensional Bravais lattices (pp. 746-747) | html | pdf |
    - Table 9.1.8.1. The 24 `Symmetrische Sorten' (p. 748) | html | pdf |
    - Table 9.1.9.1. Example (p. 749) | html | pdf |
- 9.2. Reduced bases (pp. 750-755) | html | pdf | chapter contents |
  P. M. de Wolff
  - 9.2.1. Introduction (p. 750) | html | pdf |
  - 9.2.2. Definition (p. 750) | html | pdf |
  - 9.2.3. Main conditions (pp. 750-751) | html | pdf |
  - 9.2.4. Special conditions (pp. 751-754) | html | pdf |
  - 9.2.5. Lattice characters (pp. 754-755) | html | pdf |
  - 9.2.6. Applications (p. 755) | html | pdf |
    - 9.2.6.1. Classification (p. 755) | html | pdf |
    - 9.2.6.2. Comparison of lattices (p. 755) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 9.2.4.1. The net of lattice points in the plane of the reduced basis vectors a and b; OBAD is a primitive mesh (p. 751) | html | pdf |
    - Fig. 9.2.4.2. The effect of the special conditions (p. 752) | html | pdf |
    - Fig. 9.2.4.3. The effect of the special conditions (p. 752) | html | pdf |
    - Fig. 9.2.4.4. The effect of the special conditions (p. 752) | html | pdf |
    - Fig. 9.2.4.5. The effect of the special conditions (p. 752) | html | pdf |
  - Tables
    - Table 9.2.5.1. 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. 753) | html | pdf |
    - Table 9.2.5.2. Lattice characters described by relations between conventional cell parameters (p. 754) | html | pdf |
- 9.3. Further properties of lattices (pp. 756-760) | html | pdf | chapter contents |
  B. Gruber
  - 9.3.1. Further kinds of reduced cells (p. 756) | html | pdf |
  - 9.3.2. Topological characteristic of lattice characters (pp. 756-757) | html | pdf |
  - 9.3.3. A finer division of lattices (p. 757) | html | pdf |
  - 9.3.4. Conventional cells (p. 757) | html | pdf |
  - 9.3.5. Conventional characters (pp. 757-758) | html | pdf |
  - 9.3.6. Sublattices (pp. 758-759) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 9.3.2.1. A set $[M \subset E_{2}]$ consisting of three components (p. 756) | html | pdf |
    - Fig. 9.3.2.2. A convex set in $[E_{2}]$ (p. 757) | html | pdf |
    - Fig. 9.3.6.1. Three possible decompositions of a two-dimensional lattice L into sublattices of index 2 (p. 759) | html | pdf |
  - Tables
    - Table 9.3.4.1. Conventional cells (p. 758) | html | pdf |
    - Table 9.3.5.1. Conventional characters (p. 758) | html | pdf |

Point groups and crystal classes
- 10.1. Crystallographic and noncrystallographic point groups (pp. 762-803) | html | pdf | chapter contents |
  Th. Hahn and H. Klapper
  - 10.1.1. Introduction and definitions (p. 762) | html | pdf |
  - 10.1.2. Crystallographic point groups (pp. 763-795) | html | pdf |
    - 10.1.2.1. Description of point groups (pp. 763-764) | html | pdf |
    - 10.1.2.2. Crystal and point forms (pp. 764-766) | html | pdf |
    - 10.1.2.3. Description of crystal and point forms (p. 766) | html | pdf |
    - 10.1.2.4. Notes on crystal and point forms (pp. 766-795) | html | pdf |
    - 10.1.2.5. Names and symbols of the crystal classes (p. 795) | html | pdf |
  - 10.1.3. Subgroups and supergroups of the crystallographic point groups (pp. 795-796) | html | pdf |
  - 10.1.4. Noncrystallographic point groups (pp. 796-803) | html | pdf |
    - 10.1.4.1. Description of general point groups (pp. 796-797) | html | pdf |
    - 10.1.4.2. The two icosahedral groups (pp. 797-802) | html | pdf |
    - 10.1.4.3. Sub- and supergroups of the general point groups (pp. 802-803) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 10.1.3.1. Maximal subgroups and minimal supergroups of the two-dimensional crystallographic point groups (p. 795) | html | pdf |
    - Fig. 10.1.3.2. Maximal subgroups and minimal supergroups of the three-dimensional crystallographic point groups (p. 796) | html | pdf |
    - Fig. 10.1.4.1. Subgroups and supergroups of the two-dimensional general point groups (p. 802) | html | pdf |
    - Fig. 10.1.4.2. The subgroups of the two-dimensional general point groups 16mm (4N-gonal system) and 18mm [-gonal system, including the -gonal groups] (p. 802) | html | pdf |
    - Fig. 10.1.4.3. Subgroups and supergroups of the three-dimensional general point groups (p. 803) | html | pdf |
  - Tables
    - Table 10.1.1.1. The ten two-dimensional crystallographic point groups, arranged according to crystal system (p. 762) | html | pdf |
    - Table 10.1.1.2. The 32 three-dimensional crystallographic point groups, arranged according to crystal system (cf. Chapter 2.1 ) (p. 763) | html | pdf |
    - Table 10.1.2.1. The ten two-dimensional crystallographic point groups (pp. 768-769) | html | pdf |
    - Table 10.1.2.2. The 32 three-dimensional crystallographic point groups (pp. 770-790) | html | pdf |
    - Table 10.1.2.3. The 47 crystallographic face and point forms, their names, eigensymmetries, and their occurrence in the crystallographic point groups (generating point groups) (pp. 791-793) | html | pdf |
    - Table 10.1.2.4. Names and symbols of the 32 crystal classes (p. 794) | html | pdf |
    - Table 10.1.4.1. Classes of general point groups in two dimensions (N = integer $[\geq]$ 0) (p. 798) | html | pdf |
    - Table 10.1.4.2. Classes of general point groups in three dimensions (N = integer $[\geq]$ 0) (pp. 798-799) | html | pdf |
    - Table 10.1.4.3. The two icosahedral point groups (pp. 800-801) | html | pdf |
- 10.2. Point-group symmetry and physical properties of crystals (pp. 804-808) | html | pdf | chapter contents |
  H. Klapper and Th. Hahn
  - 10.2.1. General restrictions on physical properties imposed by symmetry (p. 804) | html | pdf |
    - 10.2.1.1. Enantiomorphism, enantiomerism, chirality, dissymmetry (p. 804) | html | pdf |
    - 10.2.1.2. Polar directions, polar axes, polar point groups (p. 804) | html | pdf |
  - 10.2.2. Morphology (pp. 804-805) | html | pdf |
  - 10.2.3. Etch figures (pp. 805-806) | html | pdf |
  - 10.2.4. Optical properties (pp. 806-807) | html | pdf |
    - 10.2.4.1. Refraction (p. 806) | html | pdf |
    - 10.2.4.2. Optical activity (pp. 806-807) | html | pdf |
    - 10.2.4.3. Second-harmonic generation (SHG) (p. 807) | html | pdf |
  - 10.2.5. Pyroelectricity and ferroelectricity (p. 807) | html | pdf |
  - 10.2.6. Piezoelectricity (p. 807) | html | pdf |
  - References | html | pdf |
  - Tables
    - Table 10.2.1.1. Laue classes, noncentrosymmetric crystal classes, and the occurrence (+) of specific physical effects (p. 805) | html | pdf |
    - Table 10.2.1.2. Polar axes and nonpolar directions in the 21 noncentrosymmetric crystal classes (p. 806) | html | pdf |
    - Table 10.2.4.1. Categories of crystal systems distinguished according to the different forms of the indicatrix (p. 806) | html | pdf |

Symmetry operations
- 11.1. Point coordinates, symmetry operations and their symbols (pp. 810-811) | html | pdf | chapter contents |
  W. Fischer and E. Koch
  - 11.1.1. Coordinate triplets and symmetry operations (p. 810) | html | pdf |
  - 11.1.2. Symbols for symmetry operations (pp. 810-811) | html | pdf |
  - Tables
    - Table 11.1.2.1. Information necessary to describe symmetry operations (p. 810) | html | pdf |
- 11.2. Derivation of symbols and coordinate triplets (pp. 812-816) | html | pdf | chapter contents |
  W. Fischer and E. Koch
  - 11.2.1. Derivation of symbols for symmetry operations from coordinate triplets or matrix pairs (pp. 812-813) | html | pdf |
  - 11.2.2. Derivation of coordinate triplets from symbols for symmetry operations (pp. 813-814) | html | pdf |
  - References | html | pdf |
  - Tables
    - Table 11.2.1.1. Identification of the type of the rotation part of the symmetry operation (p. 812) | html | pdf |
    - Table 11.2.2.1. Matrices for point-group symmetry operations and orientation of corresponding symmetry elements, referred to a cubic, tetragonal, orthorhombic, monoclinic, triclinic or rhombohedral coordinate system (cf. Table 2.1.2.1 ) (p. 815) | html | pdf |
    - Table 11.2.2.2. Matrices for point-group symmetry operations and orientation of corresponding symmetry elements, referred to a hexagonal coordinate system (cf. Table 2.1.2.1 ) (p. 816) | html | pdf |

Space-group symbols and their use
- 12.1. Point-group symbols (pp. 818-820) | html | pdf | chapter contents |
  H. Burzlaff and H. Zimmermann
  - 12.1.1. Introduction (p. 818) | html | pdf |
  - 12.1.2. Schoenflies symbols (p. 818) | html | pdf |
  - 12.1.3. Shubnikov symbols (p. 818) | html | pdf |
  - 12.1.4. Hermann–Mauguin symbols (pp. 818-820) | html | pdf |
    - 12.1.4.1. Symmetry directions (p. 818) | html | pdf |
    - 12.1.4.2. Full Hermann–Mauguin symbols (pp. 818-819) | html | pdf |
    - 12.1.4.3. Short symbols and generators (pp. 819-820) | html | pdf |
  - References | html | pdf |
  - Tables
    - Table 12.1.3.1. International (Hermann–Mauguin) and Shubnikov symbols for symmetry elements (p. 818) | html | pdf |
    - Table 12.1.4.1. Representatives for the sets of lattice symmetry directions in the various crystal families (p. 819) | html | pdf |
    - Table 12.1.4.2. Point-group symbols (p. 819) | html | pdf |
- 12.2. Space-group symbols (pp. 821-822) | html | pdf | chapter contents |
  H. Burzlaff and H. Zimmermann
  - 12.2.1. Introduction (p. 821) | html | pdf |
  - 12.2.2. Schoenflies symbols (p. 821) | html | pdf |
  - 12.2.3. The role of translation parts in the Shubnikov and Hermann–Mauguin symbols (p. 821) | html | pdf |
  - 12.2.4. Shubnikov symbols (pp. 821-822) | html | pdf |
  - 12.2.5. International short symbols (p. 822) | html | pdf |
  - References | html | pdf |
  - Tables
    - Table 12.2.3.1. Symbols of glide planes in the Shubnikov and Hermann–Mauguin space-group symbols (p. 821) | html | pdf |
- 12.3. Properties of the international symbols (pp. 823-832) | html | pdf | chapter contents |
  H. Burzlaff and H. Zimmermann
  - 12.3.1. Derivation of the space group from the short symbol (p. 823) | html | pdf |
  - 12.3.2. Derivation of the full symbol from the short symbol (pp. 823-825) | html | pdf |
  - 12.3.3. Non-symbolized symmetry elements (pp. 831-832) | html | pdf |
  - 12.3.4. Standardization rules for short symbols (p. 832) | html | pdf |
  - 12.3.5. Systematic absences (p. 832) | html | pdf |
  - 12.3.6. Generalized symmetry (p. 832) | html | pdf |
  - References | html | pdf |
  - Tables
    - Table 12.3.4.1. Standard space-group symbols (pp. 824-831) | html | pdf |
- 12.4. Changes introduced in space-group symbols since 1935 (pp. 833-834) | html | pdf | chapter contents |
  H. Burzlaff and H. Zimmermann

Isomorphic subgroups of space groups
- 13.1. Isomorphic subgroups (pp. 836-842) | html | pdf | chapter contents |
  Y. Billiet and E. F. Bertaut
  - 13.1.1. Definitions (p. 836) | html | pdf |
    - 13.1.1.1. The mathematical expression of equivalence (p. 836) | html | pdf |
  - 13.1.2. Isomorphic subgroups (pp. 836-842) | html | pdf |
    - 13.1.2.1. Monoclinic, tetragonal, trigonal, hexagonal systems (pp. 836-838) | html | pdf |
      - 13.1.2.1.1. Additional restrictions (pp. 837-838) | html | pdf |
    - 13.1.2.2. Cubic and orthorhombic systems (pp. 838-839) | html | pdf |
    - 13.1.2.3. Triclinic system (p. 839) | html | pdf |
    - 13.1.2.4. Parity conditions (pp. 839-842) | html | pdf |
    - 13.1.2.5. Plane groups (p. 842) | html | pdf |
    - 13.1.2.6. Tables of matrices for isomorphic subgroups (p. 842) | html | pdf |
  - References | html | pdf |
  - Tables
    - Table 13.1.2.1. Isomorphic subgroups of the plane groups (p. 837) | html | pdf |
    - Table 13.1.2.2. Isomorphic subgroups of the space groups (pp. 838-841) | html | pdf |
- 13.2. Derivative lattices (pp. 843-844) | html | pdf | chapter contents |
  Y. Billiet and E. F. Bertaut
  - 13.2.1. Introduction (p. 843) | html | pdf |
  - 13.2.2. Construction of three-dimensional derivative lattices (pp. 843-844) | html | pdf |
  - 13.2.3. Two-dimensional derivative lattices (p. 844) | html | pdf |
  - References | html | pdf |
  - Tables
    - Table 13.2.2.1. Three-dimensional derivative lattices of indices 2 to 7 (pp. 843-844) | html | pdf |
    - Table 13.2.3.1. Two-dimensional derivative lattices of indices 2 to 7 (p. 844) | html | pdf |

Lattice complexes
- 14.1. Introduction and definition (pp. 846-847) | html | pdf | chapter contents |
  W. Fischer and E. Koch
  - 14.1.1. Introduction (p. 846) | html | pdf |
  - 14.1.2. Definition (pp. 846-847) | html | pdf |
  - References | html | pdf |
- 14.2. Symbols and properties of lattice complexes (pp. 848-872) | html | pdf | chapter contents |
  W. Fischer and E. Koch
  - 14.2.1. Reference symbols and characteristic Wyckoff positions (p. 848) | html | pdf |
  - 14.2.2. Additional properties of lattice complexes (pp. 848-849) | html | pdf |
    - 14.2.2.1. The degrees of freedom (p. 848) | html | pdf |
    - 14.2.2.2. Limiting complexes and comprehensive complexes (pp. 848-849) | html | pdf |
    - 14.2.2.3. Weissenberg complexes (p. 849) | html | pdf |
  - 14.2.3. Descriptive symbols (pp. 849-872) | html | pdf |
    - 14.2.3.1. Introduction (pp. 849-870) | html | pdf |
    - 14.2.3.2. Invariant lattice complexes (pp. 870-871) | html | pdf |
    - 14.2.3.3. Lattice complexes with degrees of freedom (pp. 871-872) | html | pdf |
    - 14.2.3.4. Properties of the descriptive symbols (p. 872) | html | pdf |
  - References | html | pdf |
  - Tables
    - Table 14.2.3.1. Plane groups: assignment of Wyckoff positions to Wyckoff sets and to lattice complexes (p. 850) | html | pdf |
    - Table 14.2.3.2. Space groups: assignment of Wyckoff positions to Wyckoff sets and to lattice complexes (pp. 851-870) | html | pdf |
    - Table 14.2.3.3. Descriptive symbols of invariant lattice complexes in their characteristic Wyckoff position (p. 870) | html | pdf |
- 14.3. Applications of the lattice-complex concept (pp. 873-876) | html | pdf | chapter contents |
  W. Fischer and E. Koch
  - 14.3.1. Geometrical properties of point configurations (p. 873) | html | pdf |
  - 14.3.2. Relations between crystal structures (p. 873) | html | pdf |
  - 14.3.3. Reflection conditions (pp. 873-874) | html | pdf |
  - 14.3.4. Phase transitions (p. 874) | html | pdf |
  - 14.3.5. Incorrect space-group assignment (p. 874) | html | pdf |
  - 14.3.6. Application of descriptive lattice-complex symbols (p. 874) | html | pdf |
  - References | html | pdf |

Normalizers of space groups and their use in crystallography
- 15.1. Introduction and definitions (p. 878) | html | pdf | chapter contents |
  E. Koch, W. Fischer and U. Müller
  - 15.1.1. Introduction (p. 878) | html | pdf |
  - 15.1.2. Definitions (p. 878) | html | pdf |
  - References | html | pdf |
- 15.2. Euclidean and affine normalizers of plane groups and space groups (pp. 879-899) | html | pdf | chapter contents |
  E. Koch, W. Fischer and U. Müller
  - 15.2.1. Euclidean normalizers of plane groups and space groups (pp. 879-882) | html | pdf |
  - 15.2.2. Affine normalizers of plane groups and space groups (pp. 882-894) | html | pdf |
  - References | html | pdf |
  - Figures
    - Fig. 15.2.1.1. 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. 879) | html | pdf |
    - Fig. 15.2.1.2. Parameter range for space groups of types C2, Pc, Cm, Cc, , , $[P2_{1}/c]$ and (p. 879) | html | pdf |
    - Fig. 15.2.1.3. Parameter range for space groups of types C2, Pc, Cm, Cc, , , $[P2_{1}/c]$ and (p. 880) | html | pdf |
    - Fig. 15.2.1.4. Parameter range for space groups of types C2, Pc, Cm, Cc, , , $[P2_{1}/c]$ and (p. 880) | html | pdf |
  - Tables
    - Table 15.2.1.1. Euclidean normalizers of the plane groups (p. 881) | html | pdf |
    - Table 15.2.1.2. Euclidean normalizers of the triclinic space groups (p. 882) | html | pdf |
    - Table 15.2.1.3. Euclidean normalizers of the monoclinic and orthorhombic space groups (pp. 883-894) | html | pdf |
    - Table 15.2.1.4. Euclidean normalizers of the tetragonal, trigonal, hexagonal and cubic space groups (pp. 895-898) | html | pdf |
    - Table 15.2.2.1. Affine normalizers of the triclinic and monoclinic space groups (p. 899) | html | pdf |
    - Table 15.2.2.2. Matrices and vectors used in Table 15.2.2.1 for the description of the affine normalizers of monoclinic and triclinic space groups (p. 899) | html | pdf |
- 15.3. Examples of the use of normalizers (pp. 900-903) | html | pdf | chapter contents |
  E. Koch and W. Fischer
  - 15.3.1. Introduction (p. 900) | html | pdf |
  - 15.3.2. Equivalent point configurations, equivalent Wyckoff positions and equivalent descriptions of crystal structures (pp. 900-901) | html | pdf |
  - 15.3.3. Equivalent lists of structure factors (pp. 901-902) | html | pdf |
  - 15.3.4. Euclidean- and affine-equivalent sub- and supergroups (pp. 902-903) | html | pdf |
  - 15.3.5. Reduction of the parameter regions to be considered for geometrical studies of point configurations (p. 903) | html | pdf |
  - References | html | pdf |
  - Tables
    - Table 15.3.3.1. Changes of structure-factor phases for the equivalent descriptions of a crystal structure in (p. 902) | html | pdf |
- 15.4. Normalizers of point groups (pp. 904-905) | html | pdf | chapter contents |
  E. Koch and W. Fischer
  - References | html | pdf |
  - Tables
    - Table 15.4.1.1. Normalizers of the two-dimensional point groups with respect to the full isometry group of the circle (p. 904) | html | pdf |
    - Table 15.4.1.2. Normalizers of the three-dimensional point groups with respect to the full isometry group of the sphere (p. 904) | html | pdf |

International Tables for CrystallographyVolume A: Space-group symmetry

Edited by Th. Hahn

Contents

International Tables for Crystallography
Volume A: Space-group symmetry