INTERNATIONAL TABLES
FOR
CRYSTALLOGRAPHY
Brief Teaching Edition of
Volume A
甘棠湖SPACE-GROUP SYMMETRY
Edited by
THEO HAHN
Contributing authors
H.Arnold:Institut fu¨r Kristallographie,Rheinisch-Westfa¨lische Technische Hochschule,Aachen, Germany.*[2,5,11]
M.I.Aroyo:Faculty of Physics,University of Sofia, bulv.J.Boucher5,1164Sofia,Bulgaria.‡[Computer production of space-group tables]
E.F.Bertaut:Laboratoire de Cristallographie,CNRS, Grenoble,France.§[4,13]
Y.Billiet:De´partement de Chimie,Faculte´des Sciences et Techniques,Universite´de Bretagne Occidentale,Brest,France.}[13]
M.J.Buerger†[2,3]
H.Burzlaff:Universita¨t Erlangen–Nu¨rnberg,Robert-Koch-Stras4a,D-91080Uttenreuth,Germany.
[9.1,12]
J.D.H.Donnay†[2]
W.Fischer:Institut fu¨r Mineralogie,Petrologie und Kristallographie,Philipps-Universita¨t,D-35032 Marburg,Germany.[2,11,14,15]
D.S.Fokkema:Rekencentrum der Rijksuniversiteit, Groningen,The Netherlands.[Computer production of space-group tables]
B.Gruber:Department of Applied Mathematics, Faculty of Mathematics and Physics,Charles University,Malostranske´na´m.25,CZ-11800 Prague1,Czech Republic.††[9.3]Th.Hahn:Institut fu¨r Kristallographie,Rheinisch-Westfa¨lische Technische Hochschule,Aachen, Germany.[1,2,10]
H.Klapper:Institut fu¨r Kristallographie,Rheinisch-Westfa¨lische Technische Hochschule,Aachen, Germany.‡‡[10]
E.Koch:Institut fu¨r Mineralogie,Petrologie und Kristallographie,Philipps-Universita¨t,D-35032 Marburg,Germany.[11,14,15]
P. B.Konstantinov:Institute for Nuclear Rearch and Nuclear Energy,72Tzarigradsko Chau, BG-1784Sofia,Bulgaria.[Computer production of space-group tables]
G.A.Langlet†[2]
stay的意思A.Looijenga-Vos:Laboratorium voor Chemische Fysica,Rijksuniversiteit Groningen,The Nether-lands.§§[2,3]
U.Mu¨ller:Fachbereich Chemie,Philipps-Universita¨t, D-35032Marburg,Germany.[15.1,15.2]
P.M.de Wolff†[2,9.2]
论美
H.Wondratschek:Institut fu¨r Kristallographie, Universita¨t,D-76128Karlsruhe,Germany.[2,8] H.Zimmermann:Institut fu¨r Angewandte Physik, Lehrstuhl fu¨r Kristallographie und Strukturphysik, Universita¨t Erlangen–Nu¨rnberg,Bismarckstras 10,D-91054Erlangen,Germany.[9.1,12]
*Prent address:Am Beulardstein22,D-52072Aachen,Germany.
‡Prent address:Departamento de Fisica de la Materia Condensada,Facultad de Ciencias,Universidad del Pais Vasco,Apartado644,48080Bilbao,Spain.
§Prent address:15rue des Moissons,F-38180Seyssins,France.
}Prent address:8place de Jonquilles,F-29860Bourg-Blanc,France.†Decead
††Prent address:Socharˇska´14,CZ-17000Prague7,Czech Republic.‡‡Prent address:Mineralogisch-Petrologisches Institut,Universita¨t Bonn, D-53115Bonn,Germany.
§§Prent address:Roland Holstlaan908,2624JK Delft,The Netherlands.
Contents
PAGE Preface to the Fifth,Revid Edition(Th.Hahn) (vii)
PART1.SYMBOLS AND TERMS USED IN THIS VOLUME (1)
1.1.Printed symbols for crystallographic items(Th.Hahn) (2)
1.1.1.Vectors,coefficients and coordinates (2)
1.1.2.Directions and planes (2)
1.1.3.Reciprocal space (2)
1.1.4.Functions (2)
1.1.5.Spaces (3)
1.1.6.Motions and matrices (3)
1.1.7.Groups (3)
1.2.Printed symbols for conventional centring types(Th.Hahn) (4)
1.2.1.Printed symbols for the conventional centring types of one-,two-and three-dimensional cells (4)
1.2.2.Notes on centred cells (4)
1.3.Printed symbols for symmetry elements(Th.Hahn) (5)
1.3.1.Printed symbols for symmetry elements and for the corresponding symmetry operations in one,two and three
dimensions (5)
1.3.2.Notes on symmetry elements and symmetry operations (6)
1.4.Graphical symbols for symmetry elements in one,two and three dimensions(Th.Hahn) (7)
1.4.1.Symmetry planes normal to the plane of projection(three dimensions)and symmetry lines in the plane of the
figure(two dimensions) (7)
1.4.2.Symmetry planes parallel to the plane of projection (7)
À3m and m3Àm only) (8)
1.4.3.Symmetry planes inclined to the plane of projection(in cubic space groups of class4
1.4.4.Notes on graphical symbols of symmetry planes (8)
1.4.5.Symmetry axes normal to the plane of projection and symmetry points in the plane of thefigure (9)
1.4.6.Symmetry axes parallel to the plane of projection (10)
描写秋天景色的成语1.4.7.Symmetry axes inclined to the plane of projection(in cubic space groups only) (10)
References (11)
PART2.GUIDE TO THE USE OF THE SPACE-GROUP TABLES (13)
2.1.Classification and coordinate systems of space groups(Th.Hahn and A.Looijenga-Vos) (14)
2.1.1.Introduction (14)
2.1.2.Space-group classification (14)
工作检讨书怎么写2.1.3.Conventional coordinate systems and cells (14)
2.2.Contents and arrangement of the tables(Th.Hahn and A.Looijenga-Vos) (17)
2.2.1.General layout (17)
2.2.2.Space groups with more than one description (17)
2.2.3.Headline (17)
2.2.4.International(Hermann–Mauguin)symbols for plane groups and space groups(cf.Chapter12.2) (18)
2.2.5.Patterson symmetry (19)
2.2.6.Space-group diagrams (20)
2.2.7.Origin (24)
2.2.8.Asymmetric unit (25)
2.2.9.Symmetry operations (26)
No.2P 1
À......................................................................
90No.4P 21(unique axis b only)..........................................................92No.12C 2/m (unique axis b only)
........................................................
94No.14P 21/c (unique axes b and c )........................................................98No.15C 2/c (unique axes b and c )........................................................106No.114No.19P 212121
....................................................................
116No.No.118No.53Pmna ......................................................................120No.62Pnma ......................................................................122No.64
Cmce (Cmca )
................................................................
124
Inside front and back covers 2.2.12.Oriented site-symmetry symbols ......................................................282.2.13.Reflection conditions ..............................................................292.2.14.Symmetry of special projections ......................................................
332.2.15.Maximal subgroups and minimal supergroups
............................................
352.2.16.Monoclinic space groups
..........................................................
382.2.17.Crystallographic groups in one dimension ................................................40References
........................................................................
41PART 3.DETERMINATION OF SPACE GROUP ..........................................
433.1.Space-group determination and diffraction symbols (A.Looijenga-Vos and M.J.Buerger)......................
443.1.1.Introduction ....................................................................443.1.2.Laue class and cell ................................................................443.1.3.Reflection conditions and diffraction symbol ..............................................
443.1.4.Deduction of possible space groups
....................................................
453.1.5.Diffraction symbols and possible space groups ..............................................463.1.6.Space-group determination by additional methods ............................................51References
........................................................................
54PART 5.TRANSFORMATIONS IN CRYSTALLOGRAPHY ..................................
555.1.Transformations of the coordinate system (unit-cell transformations)(H.Arnold)
..........................
565.1.1.Introduction ....................................................................565.1.2.Matrix notation ..................................................................565.1.3.General transformation
............................................................
565.2.Transformations of symmetry operations (motions)(H.Arnold)
......................................
645.2.1.Transformations ................................................................
645.2.2.Invariants
....................................................................
645.2.3.Example:low cristobalite and high cristobalite ..............................................65References
........................................................................
67PART 6.THE 17PLANE GROUPS (TWO-DIMENSIONAL SPACE GROUPS)
..................
69PART 7.EXAMPLES FROM THE 230SPACE GROUPS ....................................
89
No.135P 42/mbc ....................................................................130No.141I 41/amd (origin choices 1and 2)
....................................................
网上投诉怎么投诉
132No.162P 3À
1m ......................................................................136No.164P 3À
138No.166R 3À
桑葚的功效与作用m ......
................................................................
140No.194P 63/mmc ....................................................................144No.146No.205Pa 3À.........................................................
.............148No.225Fm 3À
m ......................................................................150No.227Fd 3À
m (origin choices
1and 2)......................................................
154Author index
..........................................................................
162Subject index ..........................................................................
163
Preface to the Fifth,Revid Edition
By Th.Hahn
Volume A of International Tables for Crystallography wasfirst
published in1983.Shortly after,in1985,the Brief Teaching
Edition of Volume A was prepared,of which the prent volume is
the Fifth Edition.It is bad on the Fifth,Revid Edition of
Volume A(2002).
The Teaching Edition consists of:
complete descriptions of the17plane groups,so uful for the
teaching of symmetry;
24lected space-group examples,of varying complexity and
distributed over all ven crystal systems;
tho basic text ctions of Volume A which are necessary for
the understanding and handling of space groups(Parts1,2,3
and5).
Note that space group No.64(Cmce)provides an example
containing the‘double’glide plane e.
The purpo of the Teaching Edition is threefold:
(i)It should provide a handy(and inexpensive)tool for
rearchers and students to familiarize themlves with the u of
the space-group tables in Volume A.
(ii)It is designed for u in classroom teaching,and with this
aim in mind the price has been kept as low as possible.In order to
achieve this,the material has been reprinted from Volume A
without any changes,except for pagination;hence,this Teaching
Edition contains references to ctions which are only found in
Volume A.
(iii)It may rve as a laboratory handbook becau the24
examples include most of the frequently occurring space groups,
生活中的物理for both organic and inorganic crystals.
In addition to the24space groups given explicitly,further space
groups may easily be derived by making u of the general-
position entries for the maximal subgroups of types I(translation-
engleich)and IIa(klasngleich decentred)as described in
Section2.2.15.1:The numbers given refer to tho coordinate
triplets of the general position of the group which are retained in
the maximal subgroup and thus characterize the subgroup
completely.For tho maximal subgroups which conventionally
are referred to the same basis vectors and the same origin as the
group,the‘standard description’,as given in Volume A,is
obtained.
This procedure is illustrated by the following example:
For space group No.199,I213(p.147),the following entries
are given under
Maximal non-isomorphic subgroups
I½3 I211ðI212121;24Þð1;2;3;4Þþ
which has to be read as
ð0;0;0Þþð1
2;1
2
;1
2
Þþ
ð1Þx;y;zð2Þ"xþ"y;zþ1
2
ð3Þ"x;yþ1
2;"zþ1ð4Þxþ1
2
;
"yþ1;"z:
This is identical with the general position of space group No.24,
I212121(p.217of Volume A),which is a maximal translation-
engleiche subgroup of I213of index[3].
IIa½2 P213ð198Þ1;2;3;4;5;6;7;8;9;10;11;12
which has to be read as
ð1Þx;y;zð2Þ"xþ1"y;zþ1
2
...ð12Þ"yþ1"z;xþ1
2
:
This is identical with the general position of space group No.198,
P213(p.611of Volume A),which is a maximal klasngleiche
(decentred)subgroup of I213of index[2].
(The other entries under I on p.147refer to four conjugate
maximal translationengleiche subgroups of type R3and index[4];
the entries,however,are not bad on the standard axes and
origin of R3.)
Similar relations hold for the following examples:
P"1(2)yields P1(1)
C12=m1(12)yields C121(5);C1m1(8);
P12=m1(10)
C12=c1(15)yields C1c1(9);P12=c1(13);
P121=n1(14)
Pmna(53)yields P1121=a(14);P12=n1(13);
Pmn21(31)
Cmce(Cmca)(64)yields Pbca(61)
R"3m(166)yields R32(155);R"3(148);
R3m(160);P"3m1(164)
P63=mmc(194)yields P6322(182);P63=m(176);
P63mc(186);P"3m1(164);
P"31c(163);P"62c(190)
I213(199)yields I212121(24);P213(198)
Fm"3m(225)yields Fm"3(202);F432(209);
F"43m(216);Pm"3m(221);
Pn"3m(224)
Fd"3m(227,origin1)yields Fd"3(203);F4132(210);
F"43m(216).
It is an interesting exerci to complete this list for the24
lected space groups and to extend it even to tho maximal
subgroups where the origin,the basis vectors,or both,are
different from the group;in fact,to encourage this kind of
‘playing’with space groups is one of the intentions of the
Teaching Edition.
The Editor wishes to extend his sincere thanks to the
International Union of Crystallography for making this inexpen-
sive edition possible,to D.W.Penfold,M.H.Dacombe,S.E.
Barnes and N.J.Ashcroft(Chester)for its technical preparation,
and to a number of colleagues for counl on the lection of
material,especially D.W.J.Cruickshank(Manchester)and H.
Wondratschek(Karlsruhe).
Aachen,November2001Theo Hahn
