International Tables for Crystallography
Volume A: Space-group symmetry

First online edition (2006)   ISBN: 978-0-7923-6590-7   eISBN: 978-1-4020-5406-8   doi: 10.1107/97809553602060000100

Edited by Th. Hahn

Contents

• 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

• Part 1. 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 and 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 |

• Part 2. 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 ( = general-position diagram)  (p. 21) | html | pdf |
      • Fig. 2.2.6.2. Monoclinic space groups, setting with unique axis b ( = 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 ( = general-position diagram)  (p. 23) | html | pdf |
      • Fig. 2.2.6.8. Trigonal P and hexagonal P space groups ( = 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 ( = 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 |

• Part 3. 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 |

• Part 4. 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 |

• Part 5. 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   (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   (p. 84) | html | pdf |
      • Fig. 5.1.3.8. Hexagonal lattice projected along   (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 and   (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   (p. 88) | html | pdf |

• Part 6. 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  (p. 92) | html | pdf |
    • Plane group 2  (p. 93) | html | pdf |
    • Plane group 3  (p. 94) | html | pdf |
    • Plane group 4  (p. 95) | html | pdf |
    • Plane group 5  (p. 96) | html | pdf |
    • Plane group 6  (p. 97) | html | pdf |
    • Plane group 7  (p. 98) | html | pdf |
    • Plane group 8  (p. 99) | html | pdf |
    • Plane group 9  (p. 100) | html | pdf |
    • Plane group 10  (p. 101) | html | pdf |
    • Plane group 11  (p. 102) | html | pdf |
    • Plane group 12  (p. 103) | html | pdf |
    • Plane group 13  (p. 104) | html | pdf |
    • Plane group 14  (p. 105) | html | pdf |
    • Plane group 15  (p. 106) | html | pdf |
    • Plane group 16  (p. 107) | html | pdf |
    • Plane group 17  (pp. 108-109) | html | pdf |

• Part 7. The 230 space groups
  • 7.1. The 230 space groups  (pp. 112-717) | html | | chapter contents |
    • Space group 1  (pp. 112-113) | html | pdf |
    • Space group 2  (pp. 114-115) | html | pdf |
    • Space group 3  (pp. 116-119) | html | pdf |
      • Space group 3, unique axis b  (pp. 116-117) | html | pdf |
      • Space group 3, unique axis c  (pp. 118-119) | html | pdf |
    • Space group 4  (pp. 120-123) | html | pdf |
      • Space group 4, unique axis b  (pp. 120-121) | html | pdf |
      • Space group 4, unique axis c  (pp. 122-123) | html | pdf |
    • Space group 5  (pp. 124-131) | html | pdf |
      • Space group 5, unique axis b  (pp. 124-127) | html | pdf |
      • Space group 5, unique axis c  (pp. 128-131) | html | pdf |
    • Space group 6  (pp. 132-135) | html | pdf |
      • Space group 6, unique axis b  (pp. 132-133) | html | pdf |
      • Space group 6, unique axis c  (pp. 134-135) | html | pdf |
    • Space group 7  (pp. 136-143) | html | pdf |
      • Space group 7, unique axis b  (pp. 136-139) | html | pdf |
      • Space group 7, unique axis c  (pp. 140-143) | html | pdf |
    • Space group 8  (pp. 144-151) | html | pdf |
      • Space group 8, unique axis b  (pp. 144-147) | html | pdf |
      • Space group 8, unique axis c  (pp. 148-151) | html | pdf |
    • Space group 9  (pp. 152-159) | html | pdf |
      • Space group 9, unique axis b  (pp. 152-155) | html | pdf |
      • Space group 9, unique axis c  (pp. 156-159) | html | pdf |
    • Space group 10  (pp. 160-163) | html | pdf |
      • Space group 10, unique axis b  (pp. 160-161) | html | pdf |
      • Space group 10, unique axis c  (pp. 162-163) | html | pdf |
    • Space group 11  (pp. 164-167) | html | pdf |
      • Space group 11, unique axis b  (pp. 164-165) | html | pdf |
      • Space group 11, unique axis c  (pp. 166-167) | html | pdf |
    • Space group 12  (pp. 168-175) | html | pdf |
      • Space group 12, unique axis b  (pp. 168-171) | html | pdf |
      • Space group 12, unique axis c  (pp. 172-175) | html | pdf |
    • Space group 13  (pp. 176-183) | html | pdf |
      • Space group 13, unique axis b  (pp. 176-179) | html | pdf |
      • Space group 13, unique axis c  (pp. 180-183) | html | pdf |
    • Space group 14  (pp. 184-191) | html | pdf |
      • Space group 14, unique axis b  (pp. 184-187) | html | pdf |
      • Space group 14, unique axis c  (pp. 188-191) | html | pdf |
    • Space group 15  (pp. 192-199) | html |