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

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

• 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 |

• 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 |
  • 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 |
    • 2.2.5. Patterson symmetry  (p. 19) | html | pdf |
    • 2.2.6. Space-group diagrams  (pp. 20-24) | html | pdf |
    • 2.2.7. Origin  (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 |
    • 2.2.10. 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 |
    • 2.2.14. Symmetry of special projections  (pp. 33-35) | html | pdf |
    • 2.2.15. Maximal subgroups and minimal supergroups  (pp. 35-38) | html | pdf |
    • 2.2.16. Monoclinic space groups  (pp. 38-40) | html | pdf |
    • 2.2.17. Crystallographic groups in one dimension  (p. 40) | html | pdf |
    • References | 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 |
    • References | 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 |
    • References | 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 |
  • 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 |
    • 4.3.3. Orthorhombic system  (pp. 68-71) | html | pdf |
    • 4.3.4. Tetragonal system  (pp. 71-73) | html | pdf |
    • 4.3.5. Trigonal and hexagonal systems  (pp. 73-75) | html | pdf |
    • 4.3.6. Cubic system  (pp. 75-76) | html | pdf |
    • References | 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 |
  • 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 |
    • 5.2.3. Example: low cristobalite and high cristobalite  (pp. 87-89) | html | pdf |
    • References | 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 4, P2₁  (pp. 120-123) | html | pdf |
    • Space group 5, C2  (pp. 124-131) | html | pdf |
    • Space group 6, Pm  (pp. 132-135) | html | pdf |
    • Space group 7, Pc  (pp. 136-143) | html | pdf |
    • Space group 8, Cm  (pp. 144-151) | html | pdf |
    • Space group 9, Cc  (pp. 152-159) | html | pdf |
    • Space group 10, P2/m  (pp. 160-163) | html | pdf |
    • Space group 11, P2₁/m  (pp. 164-167) | html | pdf |
    • Space group 12, C2/m  (pp. 168-175) | html | pdf |
    • Space group 13, P2/c  (pp. 176-183) | html | pdf |
    • Space group 14, P2₁/c  (pp. 184-191) | html | pdf |
    • Space group 15, C2/c  (pp. 192-199) | 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 49, Pccm  (pp. 268-269) | html | pdf |
    • Space group 50, Pban  (pp. 270-273) | 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 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 69, Fmmm  (pp. 316-318) | html | pdf |
    • Space group 70, Fddd  (pp. 320-323) | 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 86, P4₂/n  (pp. 352-355) | html | pdf |
    • Space group 87, I4/m  (pp. 356-357) | html | pdf |
    • Space group 88, I4₁/a  (pp. 358-361) | html | pdf |
    • Space group 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 126, P4/nnc  (pp. 438-441) | 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 130, P4/ncc  (pp. 450-453) | 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 134, P4₂/nnm  (pp. 462-465) | 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 138, P4₂/ncm  (pp. 474-477) | 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 142, I4₁/acd  (pp. 486-489) | 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 147, P-3  (pp. 498-499) | html | pdf |
    • Space group 148, R-3  (pp. 500-503) | 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 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 161, R3c  (pp. 532-535) | 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 167, R-3c  (pp. 548-551) | 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 202, Fm-3  (pp. 620-622) | html | pdf |
    • Space group 203, Fd-3  (pp. 623-627) | 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 223, Pm-3n  (pp. 680-682) | html | pdf |
    • Space group 224, Pn-3m  (pp. 683-687) | 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 228, Fd-3c  (pp. 704-711) | 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 |
  • 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 |
  • 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 |
    • 8.3.4. Sequence of space-group types  (p. 736) | html | pdf |
    • 8.3.5. Space-group generators  (pp. 736-738) | html | pdf |
    • 8.3.6. Normalizers of space groups  (pp. 738-739) | html | pdf |
    • References | 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 |
  • 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 |
    • References | 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 |

• 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 |
    • 10.1.3. Subgroups and supergroups of the crystallographic point groups  (pp. 795-796) | html | pdf |
    • 10.1.4. Noncrystallographic point groups  (pp. 796-803) | html | pdf |
    • References | 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 |
    • 10.2.2. 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 |
    • 10.2.5. Pyroelectricity and ferroelectricity  (p. 807) | html | pdf |
    • 10.2.6. Piezoelectricity  (p. 807) | html | pdf |
    • References | 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 |
  • 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 |

• 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 |
    • References | 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 |
  • 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 |
  • 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 |
    • 13.1.2. Isomorphic subgroups  (pp. 836-842) | html | pdf |
    • References | 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 |

• 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 |
    • 14.2.3. Descriptive symbols  (pp. 849-872) | html | pdf |
    • References | 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 |
  • 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 |
  • 15.4. Normalizers of point groups  (pp. 904-905) | html | pdf | chapter contents |
    E. Koch and W. Fischer
    • References | html | pdf |