C Standard^ 1Q73*B*3F*J Space Groups and Lattice Complexes

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ILS DEPARTMENT OF COMMERCE Nations! Bureau c Standard^ 1Q73*b*3f*J Space Groups and Lattice Complexes

Werner FISHER

Mineralogisches Institut der Philipps-Universitat D-3550 Marburg/Lahn, German Federal Republic

Hans BURZLAFF

Lehrstuhl fiir Kristallographie der Friedrich-Alexander-Universitat Erlangen-Niirnberg D-8520 Erlangen, German Federal Republic

Erwin HELLNER

Fachbereich Geowissenschaften der Philipps-Universitat D-3550 Marburg/Lahn, German Federal Republic

J. D. H. DONNAY

Department de Geologie, Universite©de Montreal CP 6128 Montreal 101, Quebec, Canada Formerly, The Johns Hopkins University, Baltimore, Maryland

Prepared under the auspices of the Commission on International Tables, International Union of Crystallography, and the Office of Standard Reference Data, National Bureau of Standards.

U.S. DEPARTMENT OF COMMERCE, Frederick B. Dent, Secretary NATIONAL BUREAU OF STANDARDS, Richard W. Roberts, D/recfor

Issued May 1973 Library of Congress Catalog Number: 73-600099

National Bureau of Standards Monograph 134

Nat. Bur. Stand. (U.S.), Monogr. 134, 184 pages (May 1973) CODEN: NBSMA6

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402 (Order by SD Catalog No. Cl3.44-.134). Price: $4.10, domestic postpaid; $3.75, GPO Bookstore Stock Number 0303-01143 Abstract The lattice complex is to the space group what the site set is to the point group - an assemblage of symmetry-related equivalent points. The symbolism introduced by Carl Hermann has been revised and extended, A total of 402 lattice complexes are derived from 67 Weissenberg com plexes. The Tables list site sets and lattice complexes in standard and alternate representations. They answer the following questions: What are the co-ordinates of the points in a given lattice complex? In which space groups can a given lattice complex occur? What are the lattice complexes that can occur in a given space group? The higher the symmetry of the crystal structures is, the more useful the lattice- complex approach should be on the road to the ultimate goal of their classification.

Keywords: Crystallography; crystal point groups; crystal structure; lattice complexes; site sets; space groups.

Acknowledgments

We wish to acknowledge support received from the Philipps- Universitat, The Johns Hopkins University, the National Bureau of Standards and the International Union of Crystallography. Special thanks go to our students and assistants, both in Marburg and in Baltimore, for their enthusiastic participation in the construction and the checking of the tables.

The authors

III T 0

THE

MEMORY

0 F

CARL HEINRICH HERMANN

(1898 - 1961)

Dr.Phil, of the University of Gottingen, Professor of Crystallography at the University of Marburg-on-the-Lahn, Germany CONTENTS Page TEXT

Introduction 1

Part I. SITE SETS 3 1. Generalities 3 1.1 Definitions 3 1.2 Rules of Splitting 5 1.3 Special Positions 6 2. Description of Site Sets 6 3. Occurrence-.s of Site Sets 9 4. Subgroup Relations 9 5. Site Sets and Their Occurrences in Oriented Subgroups 9 5.1 Cubic Degradation 9 5.2 Hexagonal Degradation 10

Part II. LATTICE COMPLEXES 11 6. Generalities 11 6.1 Definitions 11 6.2 The Lattice-Complex Symbol 12 6.21 Invariant Lattice Complexes 12 6.22 Variant Lattice Complexes 16 6.3 Rules for Selecting the Generating Lattice Complex 16 6.4 Distribution Symmetry 17 7. Weissenberg Complexes as Generating Complexes 18 7.1 The Weissenberg Complexes 18 7.2 Tables of Weissenberg Complexes 19 7.3 Figures of Weissenberg Complexes 19 8. List of the Lattice Complexes and Their Occurrences 20 9. List of Space Groups with Lattice-Complex Representations 23

Appendix. Subgroup Relations 23

References 26

FIGURES

1-47 Site Sets in Stereographic Projection 29 48 Subgroup Relations in Point Groups 34 49-53 Cubic Lattice Complexes 35 54-57 Hexagonal Lattice Complexes 40 58-60 Tetragonal Lattice Complexes 44 61-63 Orthorhombic Lattice Complexes 47 64-65 Monoclinic Lattice Complexes 50

TABLES

1 Rules of Splitting 54 Site Sets: 2 Cubic 55 3 Hexagonal and Rhombohedral 56 4 Rhombohedral (in R co-ordinates) 58 5 Tetragonal 59 6 Orthorhombic 60 7 Monoclinic 61 8 Anorthic (Triclinic) 61 9-15 Site Sets in Positions in Point Groups 62 16 Cubic Degradation 64 17 Hexagonal Degradation 66

Weissenberg Complexes 18 Cubic 69 19 Hexagonal and Rhombohedral 72 20 Rhombohedral (in R co-ordinates) 75 21 Tetragonal 76 22 Orthorhombic 79 23 Monoclinic 82 24 Anorthic (Triclinic) 83 Occurrence of Lattice Complexes 25 Cubic Invariant 85 Univariant 88 Bivariant 92 Trivariant 93 26-27 Hexagonal and Rhombohedral Invariant 95 Univariant 98 Bivariant 101 Trivariant 103 28 Tetragonal Invariant 105 Univariant 109 Bivariant 114 Tri variant 117 29 Orthorhombic Invariant 119 Univariant 121 Bivariant 124 Trivariant 127 30 Monoclinic Invariant 129 Univariant 129 Bivariant 130 Trivariant 130 31 Anorthic (Triclinic) 131

Space Groups with Lattice-Complex Representations 32 Cubic 133 33 Hexagonal and Rhombohedral 138 34 Rhombohedral (in R co-ordinates) 148 35 Tetragonal 149 36 Orthorhombic 158 37 Monoclinic 165 38 Anorthic (Triclinic) 167

Appendix

39-43 Subgroup Relations 169

Index 175

Errata 177

VI SPACE GROUPS AND LATTICE COMPLEXES

Werner Fischer, Hans Burzlaff, Erwin Heliner, and J.D.K. Donnay INTRODUCTION

A lattice is defined, sensu stricto* as the for there are as many interpentrating lattices as one assemblage of all the points that are the termini cares to pick points in the cell, whether these points of the vectors are occupied by atoms or not. As the crystal struc L(uvw] = wa + vb + we, tures that were worked out increased in complexity,the where a, S, "c are three non-cop!anar vectors and word lattice came to be used for any set of equi Uj v, w are integers that can take all possible valent atoms in a structure, whether the atoms were values, positive, negative or zero. The lattice equivalent by lattice translations or by other sym points are called nodes so they can be distinguish metry operations in the space group. Even when the ed from other points in space. A lattice is the atomic sets did occupy the nodes of lattices (e.e.), geometrical expression of a translation group: if different "lattices" were found to occur in the same a node is taken as origin, any other node can be structure; as in fluorite, for instance, where the obtained from it by means of a translation that is "calcium lattice" is cF (cubic face-centered) with one of tne vectors t(wi*j). cell edge a, whereas the "fluorine lattice", meaning As defined above, a lattice would always be the "lattice" of fluorine atoms, is cP (cubic primi primitive. In the case of symmetrical lattices, a tive) with cell edge a/2. Confusion reached a climax multiple cell can be used to define the lattice; when the word "lattice" was given another additional examples: cl, cubic body-centered; cF, cubic face- meaning to designate the collection of all the sets centered; etc. Centered lattice modes are useful of atomic centers in the crystal structure, in other in that they display the lattice symmetry, but they words, the crystal structure itself! are no more than alternate descriptions of lattices Paul Niggli (1919) introduced the term lattice that could be defined by primitive cells. Whatever complex to designate a set of crystallbgraphically description is chosen, the lattice nodes are the equivalent atoms in a crystal structure. Examples: same, and, therefore, the translation group is also the Na atoms in NaCl, the C atoms in diamond, the tne same. Cs atoms in CsCl, etc. Consider the carbon atoms After 1912 the word lattice was given various in diamond. They can be described in two ways: physical meanings, in addition to its original geo either eight "interpenetrating carbon lattices" or metrical meaning, in the description of crystal a cell content of 8 carbon atoms repeated by one structures. A structure in which all the atoms were lattice. 1 Niggli visualized the former, as was of one kind, situated at the nodes of a lattice, could common at that time among German-speaking crystallo- be described as a "lattice" of such atoms. For example graphers. Not only are the carbon atoms of any one the "lattice" of Cu atoms was used to designate the of the eight "lattices" equivalent (both crystal!o- crystal structure of copper metal. Atoms of one kind graphically and chemically) by lattice translations, were referred to as a "lattice" even in cases where but the "carbon lattices" themselves are equivalent, they did not occupy the nodes of any lattice; as the either by lattice translation or by some other space- carbon atoms in diamond, for instance, which to group-symmetry operation. The set of "atomic latt this day! -- are described as forming the "diamond ices" could thus be called a complex of equivalent lattice11 . In a crystal structure composed of more lattices or a lattice complex for short. By the than one kind of atoms, CsCl for example, as many same token a crystal form, being a set of equivalent "atomic lattices" had to be distinguished as there faces, could be dubbed a face complex. were different kinds of atoms, whence "the cesium Niggli discusses the analogy©between the crystal lattice" and "the chlorine lattice". The CsCl forms of crystal morphology and the lattice complexes structure was thus described by means of "two inter penetrating lattices" instead of "two atoms repeated 1 Note that this lattice is taken as primitive (cP); by a primitive cubic lattice". We realize nowadays with a face-centered lattice (cF), two atoms are re how unfortunate this concept of interpenetration was, peated by the lattice. of crystal structure, pointing out that this analogy In 1953, in a paper given at the Erlangen meeting is not complete. He also snows the importance of of the Crystallography Section of the German Mineral- lattice complexes 1 in three fields: (1) the des ogical Society, Hermann outlined a symbolism for uni cription of structures; (2) the crystal-structure variant lattice complexes, based on the splitting of determination itself, in which structure factors the points of an invariant lattice complex in a F(hkl) should be computed by summing over lattice number of directions depending on the available de complexes rather than individual atoms; (3) the des grees of freedom. Hermann published this talk in cription of space groups. 1960, adding new designations for the bivariant latt Niggli broaches the subject over and over ice complexes, the space-group symbol being slightly again: in the several editions of his Lehrbuch der altered by the underlining of the appropriate m. Mineralogie (1920, 1924, 1941) and in his Grundbe- P. Niggli©s original concept of lattice complex griffe (1928). Although he attempts to systematize was somewhat modified by Hermann (1935) and by J.D.H. tne nomenclature, he does not have any easy symbolism Donnay, E. Hellner and A. Niggli (1966). This last for lattice complexes. publication is a report on a Symposium held at the Carl Hermann recognized the importance of the University of Kiel, February 17-28, 1964, at which lattice complexes and devised a method for symbol the symbolism of lattice complexes was discussed by izing invariant and univariant lattice complexes, which A. de Vries and P. Th. Beurskens (University of appeared in the 1935 edition of the International Utrecht), R. Allmann, H. Burzlaff, W. Fischer, U. Tables (see Chapters VI and VII). Such a complex was Wattenberg, the late Gudrun Weiss, and the three designated by the symbol of the space group of highest rapporteurs. symmetry that contains the lattice complex as a point The fundamental feature of a lattice complexi position2 of highest symmetry; the space-group symbol which was pointed out by Hermann in 1953, is that its being followed by tne conventional Wyckoff letter of points either are themselves invariant or are disr tnis point position. If more than one such position tributed in symmetrical clusters around the (in is available, the one that includes the origin (000) variant) points of an invariant complex. Each in among its sites is the one to be chosen. This scheme variant point in its surroundings possesses its own is applicable to all lattice complexes, including the point symmetry, so that the problem largely reduces trivariant ones. to symbolizing sets of equivalent points in a point group. Part I, accordingly, deals with site sets 1 For practical applications of lattice complexes in point groups. If we draw a sphere around a point in 2 and 3 dimensions, see Burzlaff, Fischer and taken as the origin and let the symmetry elements of Hellner (1968 and 1969); Fischer (1968); Smirnova a point group go through the origin, symmetry-related (1966); Smirnova and Poteshnova (1966); Hellner points will lie on the surface of the sphere. The (1965); Loeb (1967 and 1970). poles of the faces of a crystal form constitute a 2 Point position cr, more simply, position is used familiar example of a set of equivalent points. The as a collective noun to designate a set of equivalent co-ordinates of one face pole are of the form ha*, kb* , points (either in a point group or in a space group). to* , with hj k, I integers; they are the co-ordinates It corresponds to the German Punktlage and the French of a node in the reciprocal lattice. In this work the position. The term point position was used in the co-ordinates will be taken in direct space: x - u©a, English text of the 1935 edition of the International y = v©b, z = w©c, where u 1 , v* » w f need not be Tables©, it has priority over the periphrase "set of integers. positions" used in the 1952 edition. Part II applies the results of part I to lattice complexes. PART I. SITE SETS

1. GENERALITIES valent symmetry directions. Example: in the hexa gonal system, the three columns of the symbol refer 1.1 Symbolism and definitions to: (1) the principal direction (or primary axis); The lattice, the infinite assemblage of points (2) the lateral axes of the 1st kind (Neben axes or that geometrically expresses the triple periodicity of secondary axes); and (3) the lateral axes of the 2nd a crystal, is the basis for Bravais 1 definition of the kind (zwischen axes or tertiary axes). The most sym crystal systems. It possesses, at any one of its metrical point group is the hoTohedry, 6/m 2/m 2/m or nodes, a point symmetry which is one of the 32 crystal 6/mmm for short. The symmetry directions of the lat point groups. There exist seven such lattice sym tice are vmat Carl Hermann called the Blickrichtungen metries, to which Bravais© seven crystal systems cor the directions in which you look to see the symmetry. respond. The point group of a lattice, together with Note that, in the five trigonal classes, the the subgroups that constitute its merohedries, are all Hermann-Mauguin symbol can easily be made to distin the point groups that are possible as crystal sym guish between the case of a rhombohedral lattice and metries in the corresponding crystal system. When that of an hexagonal one (Donnay, 1969): two-column the crystal symmetry is the same as the lattice sym symbols (Table 11) include 3l and 31; three-column metry, it is called holohedral symmetry or holohedry. symbols (Table 10) are explicited as 3ll and 311. In Examples: 32/m is the rhombohedral-lattice symmetry; every case the figure 1 stands for symmetry directions in the rhombohedral system it is the holohedry; 3m, of the lattice that are not symmetry directions for the 32, 3 and 3 are Its merohedries; any substance with crystal as a whole. a rhombohedral lattice can only have, as crystal sym A crystallographic site set (here called simply metry, one of these five trigonal symmetries. Tiie site set for short) is defined as a set of N equi hexagonal-lattice symmetry, on the other hand, is valent points that are symmetry related in at least 6/m 2//T7 2/m\ it has eleven merohedries (down to its one of the 32 crystal!ographic point groups and can, ogdonedry 3); any substance with a hexagonal lattice therefore, occupy a position^- in this point group. A can have, as crystal symmetry, any one of these site set will be called an ^-pointer (after the German twelve point groups, which include the five trigonal \\-Punktner or N-Punkter). Every site of the N-pointer symmetries. is invariant under certain symmetry operations, which The Hermann-Mauguin symbolism rests on the funda constitute a subgroup of order N of the crystallo mental concept of the symmetry directions in the lat graphic point group. This subgroup is called point sym tice. A symmetry direction is defined as a symmetry metry of the site or site symmetry of the point group. axis of proper or improper rotation; in other words, a A site thus lies on one, or more than one, symmetry rotation axis 1 or the normal to a mirror. Equivalent element, except in the trivial case where the site sym symmetry directions that pass through the node taken metry reduces to identity. (Symmetry elements are as the origin of a lattice, are symbolized in a col mirrors, axes and the center.) The actual symmetry umn of the Hermann-Mauguin symbol: an Arabic numeral elements in the various sites of a site set differ from indicates a symmetry axis along the direction under site to site: although equivalent mirrors and equi consideration, the letter m represents a mirror valent axes are found in every site, they are in perpendicular to it. The number of columns in the general differently oriented from one site to another. Hermann-Mauguin symbol is the number of kinds of equi- Only one point can have the center as an element of the site symmetry -- it is the fixed point in the point 1 An alternating axis (of rotatory inversion or group, taken as the origin 000. P. Niggli calls it rotatory reflection) is a symmetry direction since it Hauptsymmetriepunkt (HSP), which we propose to trans contains a rotation axis as a subgroup. Examples: 4 late by highest symmetry point. contains 2, 6 and 3 both include 3. Note that 2 is the normal to a mirror. 1 See Introduction, last footnote. Examples. A site set is provided by the corners in morphological crystallography. 1 of the cube. The eight points have the following co A limiting site set can occur only in a special ordinates 1n the usual Cartesian system, with the position in certain point groups, but not ever spe origin in the center: cial position in a given group can receive a limiting XXX > XXX, XXX©j XXX, XXX t XXX f XXX, XXX. site set (see Tables 9-15). They constitute a set of equivalent sites 1n any one In the Tables the site symmetry is given in its of three point groups: m3m (of order 48), 432 (of oriented form. In an oriented symbol dots stand for order 24) and m3 (of order 24), with site symmetry Blickrichtungen that are not used. In the hexagonal 3m (of order 6) in m3m and 3 (of order 3) in both crystal system, for instance site symmetry m may 432 and m3. In all three point groups, each site thus appear as m.., .m. , or . .m, according as the mirror lies on a 3-fold rotation axis, which coincides with is the primary mirror, one of the secondary, or one the line of intersection of three mirrors in the holo- of the tertiary mirrors. In the rhombohedral crystal hedral point group m3m. In every case the site sym system, them of the oriented symbol must always ap metry is a subgroup of index 8 of the appropriate pear in the second column of the two-column symbol point group. (.m). Another example is the assemblage of the 48 points The oriented symbol is easily recognized as the whose co-ordinates are obtained by replacing xxx 9 in Hermann-Mauguin symbol of the site symmetry, simply the first example, by each of the six complete permut by omitting the dots. By taking the dots into account, ations of xyz. These points are equivalent sites in the eTements of symmetry of the site are identified point group m3m t and their site symmetry is 1, the among those of the point group. Examples. In the trivial subgroup of index 48. tetragonal system, 2mm. indicates that the 2 is a In a point group, as a limiting case resulting subgroup of the 4, the two mirrors are the axial from a specialization of site co-ordinates, a position mirrors (represented by a single letter in the may, without change in multiplicity or site symmetry, second column of the tetragonal symbol) and the dot simulate a position that actually exists, with a dif expresses the fact that the diagonal mirrors of the ferent site symmetry,in another point group within the tetragonal symmetry are not part of this site sym same crystal system. The site set that is realized in metry. Note that 2rmt is the tetragonal way of the latter position is called the limiting site set. writing the orthornombic wn2; in both symbols the Example. In point group 23, the co-ordinates xyz may 2-fold axis is directed vertically, that is, parallel be made to obey the conditions y = *, 2 = 0; the with a. general position ixyz; then simulates the special Systems of co-ordinate axes.- The co-ordinate position :xrO: of point groups m^m and 432, In this axes used in the present Tables are those of Inter limiting case the site set in :xxQ: is the limiting national Tables (1952). In the monoclinic crystal site set that is normally realized in the other point system, we follow the 1951 Stockholm convention (b groups; it will be designated by the same site-set unique). In the orthorhombic crystal system, the symbol, which, however, will be enclosed between three columns of a symmetry symbol refer to a, 2?, c t parentheses. 1 in this order. In the tetragonal crystal system, the The site set realized in the general position smallest cell is used, so that provision must be made will be called general site set, that realized in any special position, special site set. This terminology 1 Note, however, a different usage in Int. Tab. is analogous to one that is applied to crystal forms (1935), where the crystal forms are listed on pp. 49-63: here the term "special form" is used in all 1 In Table 9 (point group 23) the parentheses around limiting cases, while "general form" applies to all I2xx recall the fact that 12xx occurs as a limiting other forms. We follow the older conventional usage: site set of 6z2xy. The multiplicity and the site general form {hkl}, a fqrm whose faces are oblique to symmetry of position:xi/J5:, where 6z2xy is realized, every element of symmetry; any other form is a special remain the same in column:xxO:/ where (12xx) occurs. form. for the two settings 42m and 4n2; the co-ordinate axes A(b). In 2/m, 4/m» 3/m(=6) and 6/m, as well as being taken along the cell edges. In the hexagonal in settings 5m2, 52m, 3ml, 3lm and 3m (which is treat crystal system, the smallest cell is used and desig ed in rhombohedral co-ordinates), the splitting point nated P (formerly called C); as in the tetragonal is made to stay in a mirror, which is either normal system, provision must be made for two possible set to, or contains, the principal axis. tings, §m2 and 62m. The rhombohedral crystal system A(c). In trie remaining point groups (Table 1, is treated twice: once in rhombohedral co-ordinates, column 1), tne point is made to move in the plane that the co-ordinate axes being taken along the lower edges passes through the origin and is perpendicular to a of the primitive cell, so that they look like the primary symmetry axis; in addition the moving point three legs of an inverted stool, and with the xr axis must retain the highest possible site symmetry, that pointing towards the observer; and again in hexagonal is to say, it must move along a secondary or tertiary co-ordinates, with a triple cell resulting from the symmetry axis, or along another primary symmetry axis so-called "R centering". The relations between the if available (along x or xx in noncubic groups, along two systems of co-ordinates are shown in International z in cubic groups). Tables (1952, vol. 1, p. 20, Fig. (a)), where the Tnis moving point reaches a site that is desig orientation of the rhombohedral axes with respect to nated by its literal co-ordinates; symmetry related the hexagonal ones is called obverse, positive or points move symmetrically and complete the assemblage direct. The additional nodes are placed at 2/3, 1/3, of equivalent sites produced by the first splitting. 1/3 and 1/3, 2/3, 2/3, so that the face (10T1) slopes B. A single polar axis. When there exists more toward the observer. than one invariant point, the fixed origin of the All our systems of co-ordinates are right-handed. point group is not completely determined by the sym metry. One invariant point is arbitrarily chosen as 1.2 Rules of Splitting. origin. Tne first "splitting" then consists in let If the point at the fixed origin is made to move ting the point move in the invariant space: in point away from it, the symmetry operations that are acting group 1, the moving point can reach any point in on it will generate N equivalent points. This is what three-dimensional space; in point group m, any point we mean when we say that the point at the origin split in the mirror; in any other hemimorphic point group, splits into the N points. If the point at the origin any point on the polar symmetry axis. is made to stay on an element of symmetry as it moves In both cases A and B, the first splitting is away from the origin, partial splitting takes place, described by the number of equivalent points, follow and further splitting will be necessary to reach com ed by the variable co-ordinates of one of the points. plete splitting, when the site symmetry of the final Examples: 2xyz, 4xz, 2xz, 4x, 6x; Ixyz, Ixz, points is reduced to identity. ly, Iz, Ixxx (Table 1). Note that, in the cubic Splitting of points is the basis of our symbolism system, in order to bring out analogies with the of site sets. It will be carried out according to the tetragonal system, the co-ordinate chosen to symbol following rules (see Table 1). ize the first splitting is 2, and the resulting six- pointer 1 is 6z. First Splitting.- Two cases are distinguished according as the point group does or does not possess Second Splitting.- If, after the first splitting, at least one homopolar axis; in the second case the the site symmetry is not the identity, the site set point group possesses a single polar axis (also called that was produced by this splitting becomes a gener heteropolar axis). ating site aet> in that its sites will serve as start A. At least one homopolar axis. In this case ing points for further splitting. there exists only a single point, taken as the origin, that is invariant under all the symmetry operations t>f 1 In crystal morphology the cube, or hexahedron, is the point group. the crystal form that corresponds to the six-pointer A(a). In point groups T, 4 and 3, complete 6z. The Dana System of Mineralogy symbolizes the splitting is immediate; the first splitting yields, form as {001} in preference to {100}, which is the respectively, the 2, 4 and 6 sites of the general symbol used by many other authors. Similar problems position. call for similar solutions I A. If the site symmetry contains a single element 2.mm, 4.,, 2..); 4xx (m.m2, ..2); in the rhombohedral (m, 2, 3, 4 or 6), only one more splitting, b/ means point groups, 2xxx (3m, 3.), 6xx (.2). Note that the of this element, is needed to bring the site symmetry hemimorphic symmetries, 2, 4, 3, 6, mm2, 4mm, 3m, 6mm, down to 1. are given their oriented symbols in the above list. B. In every remaining case, the site symmetry Other special positions have two degrees of free consists of n mirrors intersecting in an n-fold axis. dom and, in them, the site symmetry goes down to m, If the mirrors are of one kind, the second splitting either ,.m in 6z4xx (cubic), Iz4xx, 4xx2z (tetra takes place in these mirrors (example: 3xx in 3ml, gonal), 6x2z and 3x2z (hexagonal); or .m. in 4x2z but 3x in 31m). If the mirrors are of two kinds, the (tetragonal), Iz6xx, 6xx2z and 3xx2z (hexagonal). point is made to move in the mirror that contains the Note that the letters z, x, y that occur in a next co-ordinate axis, in the cyclic sequence (exam site-set symbol express successive degrees of freedom, ple: splitting 2x follows splitting Iz in mm2). In as opposed to co-ordinates of points. Note also, case B, a third splitting is necessary to reduce the however, that points on the Zwischen axes are sym site symmetry to 1. bolized 3xx, where the letter x indicates that the first degree of freedom is being used and xx indi Third Splitting.- When the generating point still cates that the second co-ordinate is negative and has lies in a mirror after the second splitting, the third the same numerical value as the first one. splitting takes it off the mirror, by making use of the third co-ordinate. Although the last splitting is 2. DESCRIPTION OF SITE SETS taking place astride the mirror, the two resulting points need not lie on a line perpendicular to the The site sets occur in the 32 crystallographic mirror. This V-shaped splitting simplifies the use point groups as assemblages of equivalent points. of the third co-ordinate. The site symmetry is now These symmetry-related points lie on a sphere around reduced to 1 (Table 1). the origin. If they are considered to be the poles The final site-set symbol, after complete split of crystal faces, the assemblage is a representation, ting, is given (column 9, Table 1) for each of the 32 in spherical projection, of a crystal form. The lines crystal!ographic point groups and for both settings radiating from the origin to the face poles would wherever two settings are possible. The rhombohedral thus be face normals. If, on the other hand, the sym co-ordinates (Miller axes) are provided for, as well metry-related points on the sphere are considered to as the hexagonal ones (Bravais axes), in trigonal be the vertices of a polyhedron, the assemblage be point groups. comes a representation of the co-ordination around the origin. In the polyhedron that idealizes a crystal 1.3 Special Positions form, the faces are equivalent, and the polyhedron is In the process of deriving the site-set symbols called isohedral\ in the co-ordination polyhedron the for the general positions in the crystal 1ographic vertices are equivalent, and the polyhedron is called point groups, a number of special positions have been ieogonal (referring to the solid angles at the ver encountered as intermediate steps. Additional special tices). The two polyhedra are thus dual of each other. positions occur in certain point groups. It follows that we have two ways of describing Some of the additional special positions possess co-ordination of sites around the origin: (1) the one degree of freedom and, in them, the site symmetry co-ordination polyhedron of the chemists; (2) the reduces to a hemimorphic subgroup of the point group. dual polyhedron, whose face normals are the equivalent The symbols of the corresponding site set are given elements. here, each one followed by the oriented symbols 1 of The site sets are shown in stereographic projec its possible site symmetries, enclosed between paren tions (Figs. 1-47); they are classified on the basis theses: in the cubic point groups, 8xxx (.3m, .3.), of their multiplicity. A site set of multiplicity N 4xxx (.3m, .3.), 12xx (m.m2, ..2); 1n the hexagonal is called an ^-pointer. It is easy to see how the point groups, 2z (6mm, 3.m, 3m. t 6.., 3..), 6xx (w?£, crystal-form polyhedron gives rise to the co-ordina ..2); in the tetragonal point groups, 2z (4mm, 2im., tion polyhedron and to coin a name or provide a des cription for the latter. 1 See Sn. 1.1. 2.1 The One-pointer.- The only one-pointer 1s the and their duals are as follows: TRUNCATED TETRAGONS, MINT (F1g. 1), corresponding to the t&NQHEDRON (or coplanar with 0 (Fig. 21) or not (Fig, 22) (ZOTSTJW- pedion). GONAL PRISM and DITETRAGQMl PYRAMID) * three TETRA GONAL site sets that are described by combinations of 2.2 The Two-pointers.- The two two-pointers are the TETRAHEDRON (Fig. 23) *PRISM (Fig. 24), and TRAPSZO- LIKE SEGMENTS* one of wnicii is col linear with the HEDROK (Fig, 25), each with the PIMCOID* correspond origin (Fig, 2) and corresponds to the PINACOID (or ing to TETRAGONAL SCALENOHEDRON* TETRAGONAL DIPYRAMID parallelehedron), while the other is not collinear with and TETRAGONAL TRAPEZOHEDRON* respectively; theVMCT- the origin (F1g. 3) and corresponds to the DIHEDRON1 . ANGULAR PARALLELEPIPED (Fig. 26)(RHOMBIC DIPYRAMID) ©* the CUBE (OCTAHEDRON) (Fig. 27). 2.3 The Three-pointers.- The two three-pointers are the TRIGONS (or equilateral triangles), one coplanar 2.7 The Twelve-pointers.- Ten twelve-pointers, six with the origin (Fig. 4), the other not (Fig. 5). of which belong to the hexagonal system and four to Their duals are, respectively, the TRIGONAL PRISM and the cubic system, have the following designations, trie TRIGONAL PYRAMID. The names of the duals are enclosed in parentheses. TRUNCATED HEXAGON, coplanar with 0 (Fig, 28) (DIHEXA- 2.4 The Four-pointers.- There exist seven four- GONAL PRISM) or not coplanar with 0 (Fig. 29) (DIUEXA- pointers: two RECTANGLES* one coplanar with the origin GONAL PYRAMID)* PINACOID WITH RHOMBOHEDRON (Fig. 30) (Fig. 6} (dual , RHOMBIC PRISM)* the other not (Fig.7) (HEXAGONAL SCALENOHEDRON) * TRUNCATED TRIGONAL PRISM (dutf*RHOMBic PYRAMID) \ two TETRAGONS (or squares), WITH PINACOID (Fig. 31) (DITRIGONAL DIPYRAMID)* HEXA likewise coplanar with 0 (Fig. 8) or not (Fig. 9) GONAL PRISM WITH PINACOID (Fig. 32) (HEXAGONAL DI (duals: TETRAGONAL PRISM and TETRAGONAL PYRAMID, res PYRAMID), HEXAGONAL TRAPEZOHEDRON WITH PIMCOID (Fig. pectively); and three tetrahedra, which have the same 33) (HEXAGONAL TRAPEZOHEDRON)* TRUNCATED TETRAHEDRON names as their duals: RHOMBIC TETRAHEDRON (or dis- (Fig. 34) (TRIGON-TRITETRAHEDRON) 1 ; CUBE WITH TWO phenoid) (Fig. 10) , TETRAGONAL TETRAHEDRON (or d1s- TETRAHEDRA (Fig. 35) (TETRAGON-TRITETRAHEDRON) 1 ; phenoi d) (Fig. 11), and REGULAR TETRAHEDRON (Fig. 12). CUBOCTAHEDRON (Fig. 36) (RHOMB-DODECAHEDRON)©* PENTAGON- TRITETRAHEDRON WITH TWO TETRAHEDRA (Fig. 37) (PENTAGON- 2.5 The Six-pointers.- Eight six-pointers comprise: TRITETRAHEDRON) ; DIHEXAHEDRON WITH OCTAHEDRON (F1g. 38) two TRUNCATED TRIGONS* coplanar with 0 (Fig. 13) (dual, (DIHEXAHEDRON or PENTAGON-DODECAHEDRON) . DITRIGONAL PRISM) or not (Fig. 14) (dual, DITRIGONAL PYRAMID)* two HEXAGONS, coplanar with 0 (Fig. 15) or 2.8 The Sixteen-pointer.- The only sixteen-pointer not (Fig. 16), corresponding to the HEXAGONAL PRISM is a combination of the TRUNCATED TETRAGONAL PRISM and HEXAGONAL PYRAMID* respectively; the OCTAHEDRON WITH THE PINACOID (Fig. 39), whose dual is the DI- (Fig. 17), whose dual is the CUBE (or hexahedron); the TETRAGONAL DIPYRAMID. TRIGONAL ANTIPRISM (Fig. 18), whose dual is the RHOM-. BOHEDRON (in crystal-form language, a trigonal anti- 2.9 The 24-pointers.- There exist six 24-po1nters. prism is described as the combination of a rhombo- They are listed here with the dual polyhedron shown nedron with the pinacoid); and two more co-ordination between parentheses: TRUNCATED HEXAGONAL PRISM WITH polyhedra that are best described as combinations of PINACOID (Fig. 40) (DIHEXAGONAL DIPYRAMID)* TRUNCATED forms: TRIGONAL PRISM WITH PINACOID (Fig. 19) and CUBE (Fig. 41) (TRIGON-TRIOCTAHEDRON) 2 * RHOM&ICUBOC- TRIGONAL TRAPEZOHEDRON WITH PINACOID (Fig. 20), whose TAHEDRON (Fig. 42) (TETRAGON-TRIOCTAHEDHON) 2 ; CUBE duals are the TRIGONAL DIPYRAMID and TRIGONAL TRAPE WITH TWO TETRAHEDRA (Fig. 43) (HEXATETRAH8DRON) ©* ZOHEDRON > respecti vely. TRUNCATED OCTAHEDRON (Fig. 44) (TETRAHEXAHEDRON) ; CUBE WITH OCTAHEDRON AND DIHEXAHEDRON (Fig. 45) 2.6 The Eight-pointers.- The seven eight-pointers (DIDODECAHEDRON), SNUB CUBE (Fig. 46) (PENTAGON-TRI- OCTAHEDRON). 1 The single term dihedron replaces the two terras used by Groth, dome and sphenoid. A crystal form 1 The two polyhedra belong to the same site set should have the same name regardless of the face sym 6z2xx (Table 2). metry (Boldyrev, 1925). 2 Two two polyhedra belong to the same site set 6z4xx (Table 2). 2.10 The 48-pointer.- The only 48-pointer is the hedra 1 , especially as regards the nomenclature of the TRUNCATED CUBOCTAHEDRON (Fig. 47), whose dual is the polyhedra and the occurrence of the site sets in point HEXAOCTAHEDRON. groups. For additional data, such as topological and Note that, in the above nomenclature, a truncated metrical site-set symbols, and limiting cases, this polygon is symmetrically truncated on the angles, a Report should be consulted. truncated polyhedron (tetrahedron, cube, octahedron) The following remarks will help in the under is likewise symmetrically truncated on the corners, standing of the tabular presentation. but a prism, being an open form with no vertices, is Condensed description of the site co-ordinates.- truncated on its edges. The rhombicuboctahedron re Instead of listing the co-ordinates of all the equi sults from the combination of a cube with octahedron valent sites in the position, it may be advantageous and rnomb-dodecahedron. The snub cube is a cube modi to use a scheme of condensation. One triplet of co fied by octahedron and pentagon-trioctahedron. The ordinates specifies one of the sites of the position, truncated cuboctahedron shows the twelve rhomb-dode- various signs explain how the other sites are derived cahedral faces truncating the twelve corners of the from the first. cuboctahedron. The Greek letter kappa ( K ) stands for the initial The site sets that occur in the 32 crystallo- of kyklos, circle, and indicates the three cyclic per graphic point groups are listed in Tables 2 to 8, mutations of the preceding triplet or of two (or more) limiting cases not included. Under a subheading con triplets enclosed between parentheses. sisting of the names of the co-ordination polyhedron The Greek letter pi U) indicates the six com and dual crystal form, the symbol of the site set is plete permutations of the preceding triplet. followed by the co-ordinates of the equivalent sites, The Greek letter tau (T) is used to signify that expressed in a condensed manner (see below), and the the first two co-ordinates must be interchanged. symbols of the positions where the site set occurs. In the hexagonal system when three co-ordinates The position is given by the point-group symbol and are used (a,, a«, c axes) (in contradistinction to the position letter between parentheses. This letter Weber©s four co-ordinates), the Greek letter iota (i) is taken from Carl Hermann©s description of the point indicates the following operations: (xyz)\ = xyz-, groups (so-called crystal classes) 1 . y, x-y , z\ y-x, x, z. (They correspond to the cyclic The multiplicity of each site set can be obtained permutations of the first three co-ordinates in a four- immediately by making the product of the numerals that co-ordinate symbol in which the third co-ordinate is indicate splitting in the site-set symbol. Example: zero: xyO; Oxy = 0-yf x-y, y-y; yOx = y-x, 0-x, x-x. 6z2x has multiplicity 12.© No explicit mention of the The sign, which should be read as "plus and multiplicity is made in the Tables. A given site set minus", is the expression of centrosynmetry; it may occur in different positions, in the same point doubles the number of equivalent sites. The sign group or in different point groups, and can be de may precede all other statements or a single letter scribed by different symbols, which will be found (as in x, y t z}. listed under one another, together with the co-ordi Triplets of signs (example: +--) can be followed nates of the sites and the positions of occurrence. by a Greek letter, with the same obvious meaning. When a site-set symbol corresponds to two names, these Example: the last site set in Table 2 is the 48- are listed under each other in the subheading. Alter pointer; the 48 triplets of co-ordinates that would be nate orientations of point groups (4m2, 42/7?, for in necessary to give all the co-ordinates explicitly can stance) have been taken into consideration. The five be condensed into the statement trigonal point groups are listed twice: once in rhom- +(+++, +--<) (xyztt)* bohedral co-ordinates (Miller axes) under rhombohedral which is easily deciphered as follows: take the six system and again in hexagonal co-ordinates (Bravais complete permutations of the triplet xyz t first with axes) under hexagonal system. the signs +++ and then with the three cyclic per The above presentation is largely based on the mutations of +--; double the number of points, from Report of the Kiel Symposium on Co-ordination Poly- 24 to 48, by changing all the signs. Another way of

1 Chapter III in Int. Tab. (1935). J.D.H. Donnay, E. Hellner and A. Niggli (1964). condensing would be (x, y, z\ x 9 y> 2)*, which is {001} and tne lower monohedron {001}. This is the just as easily interpreted. We consider such state reason why two positions are listed in such a case. ments helpful in evoking the picture of the assemblage Example: 4rm(a) and Arm(b) (Table 5). of points. Stimulating and interpretative, as well as space-saving, they are preferable to the bland catalog, 3. OCCURRENCES OF SITE SETS xyz yzx zxy yxz xzy zyx xyz yzx zxy yxz xzy zyx Site sets occur in positions in point groups. For xyz yzx zxy yxz xzy zyx each crystal system (Tables 9-15), we give a double xyz yzx zxy yxz xzy zyx entry tabulation: each row stands for one point group, xyz yzx zxy yxz xzy zyx each column for a type of co-ordinate triplet. At the xyz yzx zxy yxz xzy zyx intersection of row and column, the symbol of the site xyz yzx zxy yxz xzy zyx set 1 appears over the corresponding site symmetry of xyz yzx zxy yxz xzy zyx, the occupied position. This site symmetry is sym which dulls the senses and discourages visualization. bolized in the format of the point-group symbol, that In the description of hexagonal site sets, and of is to say that it is oriented with respect to the rhombohedral ones in hexagonal co-ordinates, we re symmetry directions of the crystal lattice (see commend the elegant Weber four-index symbol xyjz for Sn. 1.1). the co-ordinates of points. All four Bravais axes must be used: the third co-ordinate, j, which is made 4. SUBGROUP RELATIONS equal to the negative sum of the first two, is not superfluous: it cannot be left out at will, as can The relations between groups and subgroups among the i index in a face symbol (hkil} . The advantage of the 32 crystal!ographic point groups have been pre these symbols (Weber, 1922; Donnay, 1947a, b) is that sented, in the form of a line diagram, by Carl permutations of the first three co-ordinates are made Hermann in the 1935 edition of the International available to express the symmetry of the point assem Tables. blage. In fact, the parallelism between the presen This diagram is reproduced here as Fig. 48. tation of the co-ordinates xyjz of equivalent points and that of the indices hkil of equivalent faces is 5. SITE SETS AND THEIR OCCURRENCES so perfect that the two symbols are interchangeable. IN ORIENTED SUBGROUPS This property is the consequence of the mathematical symmetry of tne equation of zone control in four-index 5.1 Cubic Degradation (Table 16).- symbolism, The top lines show the site sets and the positions hx + ky + ij + Iz = 0, in the cubic point groups. Going down in the first which expresses the condition for the face (hkil} to column, we list the various subgroups in order of de be contained in the zone [xyjz©]. creasing symmetry. The tetragonal point groups, for For calculations, however, symbols with three co instance, are listed in the first column in their ordinates are more convenient. For this reason they cubic orientation; they are repeated in the second are used in Table 3. The transformation from four- column in their standard tetragonal notation. index to three-index symbols is immediate; it consists Further down the Table, the trigonal point groups in subtracting the third co-ordinate j from each of are likewise shown twice: as subgroups of the cubic the first three, so that xyjz becomes x- j , y- j , groups, in cubic orientation, in the first column and (0,) z. Example: 5T42 gives 5-5, T-4, 2 = 932. Con as standard trigonal point groups, in the third versely, transforming from three to four co-ordinates: column. 932 gives 9-(9+3)/3, 3-(9+3)/3, -(9+3)/3, 2 = 5T?2. The orthorhombic point groups appear three times: It may not be superfluous to stress that any co in the first column, as oriented subgroups of the ordinate can be assigned negative or positive values. cubic symmetries; in the second column, as oriented It follows that the single point of the site set Iz, which may have a negative as well as a positive value 1 Or the symbol of the limiting site set, which is for z in the co-ordinate triplet OOz, corresponds to written between parentheses where it appears as a two crystal forms in morphology: the upper monohedron limiting case (see Sn. 1.1). tetragonal subgroups; and, in the fourth column, as 5.2 Hexagonal Degradation (Table 17).- orthorhombic point groups in their own orientation. The subgroups of the hexagonal holohedry are listed, The same scheme is carried out all the way down to including the orthorhombic, monoclinic, and triclinic the triclinic point groups. point groups.

10 PART II. LATTICE COMPLEXES

6. GENERALITIES complex, either in a given space group or in different space groups, may be differently oriented and their 6.1 Definitions.- site symmetries may also be different. Such positions A configuration means a set of n specified 1 con constitute different representations of the lattice gruent primitive lattices, whose nodes are symmetry- complex. related sites in at least one of the 230 space groups. One of the representations of a lattice complex, Every one of the n lattices may be represented by one in tne space group where it shows the highest site node that is a site in the conventionally chosen cell. symmetry, is cnosen as the standard representation. Every site in the configuration is invariant under Our choice follows, in general, that already made by certain symmetry operations, whicn constitute a sub Carl Hermann (Int. Tab., 1935). The multiplicity of group of the point group to which the space group be a lattice complex is the number of its equivalent longs. The subgroup, which may be the trivial sub points comprised within the conventional cell of its group 1, is called point symmetry of the site or site standard representation. symmetry. A site may happen to lie on an element of General lattice complexes are trivariant; special symmetry whose operation involves a translation com lattice complexes can be bivariant, univariant or in ponent (a screw axis or a glide plane of symmetry), variant, If a position contains all the configurations but such an element cannot be part of the site sym of a second position, but the second position does not metry, which by definition is a point group. contain all the configurations of the first, then the In a given space group, two configurations are lattice complex realized in the second position will said to occupy one and the same point position (or be a limiting lattice complex (LLC) of the lattice com position for short) if, to every site in the one con plex of the first position. The subset of configura figuration, there corresponds, in the other config tions (in the first position) that coincide with the uration, a site that lies on the very same elements of set of configurations of the limiting lattice complex site symmetry. will simulate this complex as a limiting case. The condition for two positions, in tiie same Example. In space group 23 the lattice complex that space group or in different space groups, to belong to is realized in position (j) \xyz\ possesses a subset the same lattice complex (or complex for short) is of configurations that coincides with the set of con that, to every configuration of the first position, figurations of another complex, which is realized in there corresponds, in the second position, a congruent positions (i) ixxQ: of space groups Pm3m and P432. or enantiomorphous, configuration -- and conversely. In space group ^23, position (j), for y = x, z = 0, As a consequence of this definition, all the positions simulates position (i) of Pm3m and p432, while retain in which a lattice complex can be realized occur in ing its own multiplicity and site symmetry. one and the same crystal system. 2 The lattice complex of only one special position, The positions that belong to the same lattice namely, position 48(/):xO^: in laid is not a limiting lattice complex: it cannot occur as limiting case in 1 Specified in the sense that the co-ordinates of any position of any space group. All other special the nodes have specified numerical values, in any con lattice complexes are limiting lattice complexes. venient system of absolute co-ordinates. The space group in which a lattice complex is The two lattice complexes tP and cP are distinct defined in its standard (or normal) representation is complexes because tP (class of all the sets of points called the characteristic space group of the lattice that occupy the nodes of a tetragonal primitive lat complex. Other representations for which the site tice) contains, in addition to all the configurations symmetry is equal can only occur in the same charac of cP, configurations© that are not included in cP, teristic space group; other representations, with i.e. configurations of points that do not occupy the lower site symmetry, may be found in another space nodes of a cubic primitive lattice. group or in several other space groups, as well as in

501-563 0-73-2 the characteristic space group; nonstandard represen which remains invariant in spite of this choice of tations will be referred to as alternate represen origin. tations. In the monoclinic system (Table 26) P and C are invariant lattice complexes in standard orientations. 6.2 The Lattice-Complex Symbol.- Among the lattice-complex letters, we find all the 6.21 Invariant Lattice Complexes.- An invariant letters that are used to characterize lattice center lattice complex, that is to say, a complex that can ings, since the nodes of any lattice, according to occupy a parameterless position in a space group, is definition, are the equivalent points of a lattice designated by a short symbol, which consists in a complex. single letter when referring to the standard represen Some of the symbols listed (Table A) for invar tation. Example: in the anorthic (triclinic) system iant lattice complexes are topological symbols, which (Table 25), P is the symbol of an invariant lattice means that they can be used in more than one crystal complex in standard representation, which occurs in system. The symbol must be expanded in order to space group PT as position (a), with site symmetry T specify the system in which the complex is to be con and multiplicity 1. PI is its characteristic space sidered. This is achieved simply by using the ini group. tial of the name of the system as a prefix to the The same lattice complex occurs in an alternate lattice-complex symbol. Examples: cP, hP, rP, tP, representation in PI(a), with site symmetry 1. In oP, mP, aP. Note that t stands for tetragonal, r this space group the complex occupies a general for rhombohedral, a for anorthic (which is used in position :xyz: and is symbolized P[xyz], The co stead of trielinic in order to avoid conflict between ordinates are placed between brackets to indicate initials). The system need not be specified, of that they represent another origin for the complex P, course, unless there is danger of confusion.

12 ?

,, 16

. 24

3 16

3 8

v Y-l

Q©l

R-2

111

- "

= ~

v Y

©^000,333,333)

©R

;

v Y

111

222

v Y

cell,

hR;

753 885"

3 K

771

8S8

hexagonal

137

cell,

HI 824*

245©

551

585"©

S**

333 ? 452©

2<;3

"Y*

511 "V

331 524©

333, 33 753 45 855

Os=,

558

O-T,

©S

+ +

rhombohedral

V Y

111

+ Hi Ho,

112111 sOO, + +

n 45©

S 48

2 W

000, 585©

888©

v Y

Q ^

Y**

R +

, W*

+w W

T T

00*M

M tC

Multiplicity

and

Examples

©M

representations

Examples

cell

cell)

Definition,

(rJ)

cell)

hexagonal

cell

equivalent

331

(hexagonal

cell

Complexes,

0lCt

the

(hexagonal (hexagonal

2jO

oil

l^l

213 HI HQ<

21 35

334

rhombohedra

0^0,

Lattice

Hl^

repeated

Co-ordinates

points

ooo,

12 121

nnn ooo, V*

33 ©

ooo, ooo,

jOO,

000

Invariant

L.C.

Table

Symbol

C A D

I F

M P Distinction between a lattice complex and its re obtained from R by a 180 rotation around the c axis presentations.- We may distinguish between the con (in hexagonal co-ordinates); ©S is derived from S by cept of the lattice complex and the various represen a screw rotation of power 1 or 3 around a 4] axis. It tations that the complex can have in one space group is true that ©S can also result from S by an inversion or in different space groups. The standard represen through the origin (or the center of the cell), but tation will be designated by the symbol of the lat this inversion can always be combined with another tice complex itself. The symbols of all the invariant operation of the 2nd kind that is a symmetry operation lattice complexes in their standard representations of the complex to produce an operation of the 1st kind. are listed alphabetically in Table A (column 1). To obtain a minus representation from the plus repre Examples of alternate representations will be found in sentation, on the other hand, the only operation column 3. The co-ordinates of the equivalent points available is one of the 2nd kind, the inversion in the conventional cell (column 2) and the multi through the origin, for instance. plicity m (column 4) complete the Table. Other alternate representations can be obtained The symbols used for the invariant lattice com by changing the scale in certain directions (not nec plexes are essentially a typographical simplification essarily all directions), in such a way that the crys of tnose that were proposed by Carl Hermann (1960). tal system is preserved. The transformation matrix The new symbols were agreed upon at the Kiel Symposium that expresses the cell edges a©, £©, c 1 in terms of (Donnay et al. t 1966). They can be remembered without the edges a, £, c, of the complex in standard repre too much difficulty, as each letter chosen recalls sentation is given in Tables 25-31. Where no matrix some structural feature of the complex. When two re is listed, unit matrix is to be understood. presentations of a lattice complex of multiplicity m Examples: P, (Table 26) is a representation of P combine to form a new complex with multiplicity 2m, with b - b©/2 9 so that two standard representations of the complex letter is usually 1 retained but an asterisk the complex occupy the volume of the cell. The trans follows it in superscript. Example: J* (pronounce: formation matrix 1 that gives the cell edges from the J star}. If the process is repeated, the complex of edges of the complex is 100/020/001. multiplicity 4 still has the same letter, this time A small letter in subscript thus indicates that with a double asterisk. Whenever a complex possesses the corresponding edge of the complex is to be multi two enantiomorphous representations, the standard re plied by 2 in order to give the length of the cell of presentation is symbolized by a letter preceded by a the lattice in which the complex is represented. Two plus sign in superscript. Example: Q. The represen small letters in subscript, as in P , , mean that both tation chosen as standard is the one that contains a and b are to be doubled to give the dimensions of left-handed screw axes. In this convention, the sign the cell. If all three edges are to be doubled, the placed in superscript is that of the optical activity. subscript used is 2. P2 , for instance (Table 31), is Alternate representations can be enantiomorphous a representation in which the cell volume contains 8 or congruent to the standard orientation. The minus times the P complex in standard representation. The representation of a plus complex will carry the minus transformation matrix "complex to cell" is 200/020/002. sign in superscript in front of the letter. A con Examples of representations that result from a gruent representation that is obtained from the stand rotation combined with a contraction or an expansion ard representation by a translation will retain the of a mesh are found in the tetragonal system (Table same letter, but it will be preceded by the shift 28) and in the hexagonal system (Table 29-30). Con vector expressed by its three components in fractions sider a tetragonal F cell, with its 4 nodes. It has of the cell edges. Examples: **«J is a shifted re- twice the volume of an I cell, which is oriented at 111 333 presentation of J; ^F and jjjj|F are shifted represen 45 to it. The standard representation of the lattice tations of F. All other congruent representations complex is I, with multiplicity 2. The alternate re will have their symbolic letter of the standard repre presentation, in the F cell, has multiplicity 4. The sentation preceded by a prime sign. Examples: ©R is F cell is obtained from the cell of the standard com plex by transformation 110/110/001. It is rather in 1 Examples of exceptions: F + --- F is not de felicitous to describe this transformation as a 45 signated F* , it is called ?2 instead (see below); P + P is I, not P*. 1 See Sn. 8.1, footnote.

14 counterclockwise rotation of the mesh accompanied by P + P© = I -(note tnat I is preferred to P*), an expansion of the lengths in the ratio 1 to /2. J + J 1 = J*, While it is true that, if the Q length is kept con F + F 1 = ?2 (note that F* is not used), stant, this peculiar operation will transform the P + J = F, P© + J 1 = F 1 , small cell into the large one, it is also true that = P2» the same operation will not turn an I cell into an F F + F" = D, F 1 + F 1 " = D©, cell! The symbols proposed by Carl Hermann make use hP + G = hP VJ hP + N = hP. of the letters v and V (from the Latin vertere, to turn) to indicate the 45 "turn" in the tetragonal Explanation of the symbolic letters used for in system and the 30 "turn" in the hexagonal system, variant lattice complexes.- The letters that also respectively. We did not feel it would be justified describe the variously centered cells of the Bravais to modify this symbolism, misleading though it might lattices are, of course, self-explanatory. The other be. letters recall some structural feature of the complex, In the above example, the I complex in the F re as follows: presentation could have been designated I . Likewise D Diamond structure. a P complex in the C representation could have been E Hexagonal close packing (Italian- esagonale). called P . (The familiar lattice letters F and C have Tne i©lg atoms in the ilg structure occupy the equi been retained for the sake of convenience, even though valent points of the E lattice complex. ! the unity of the symbolism is thereby impaired.) G Graphite structural layer. The B atoms in AlB^ Consider now a D complex in a tetragonal cell, occupy the G complex. that is, a tD complex. (In the cubic system, cD is J Defined in standard representation as 3 points in exemplified by the diamond structure.) The tetragonal the midpoints of the cell faces. Consider the cell in question will be an F cell, which we will want six half-points within the cell and connect oppo to describe in the conventional I setting. The trans site half-points: the resulting figure has the formation matrix lio/iio/001 expresses the 45 clock shape of the American toy called "jack". (The wise rotation with shrinkage of the cell. The repre center point of the jack is the center of the sentation of the D complex in the small cell will have cell, Jij.) The atoms of oxygen in perovskite multiplicity 4, instead of 8, and the complex symbol occupy the J complex. will be V M Complex that results from the repetition of the A mnemonic rule may be found useful: to go to a equipoints of N by a rhombohedral lattice, in a larger cell, the co-ordinate axes a, and a~ are ro hexagonal cell. The Ni atoms in Nl-Pt^S^ occupy tated counterclockwise, that is, to the right, and the the M complex. subscript v is placed to the right of the letter; to N Net. Three points per mesh, extending into a go to a smaller cell, the co-ordinate axes a-j and a^ two-dimensional 4-connected net, are repeated by are rotated clockwise, that is, to the left, and the the hexagonal lattice. The Co atoms in CoSi superscript v is placed to the left of the letter. illustrate the N complex. The same rule applies to the hexagonal case, where the +Q Quartz. The Si atoms in high-quartz occupy the capital letter V is used for the 30 turn. Q complex. In the hexagonal system (Table 29-30) one addi S This complex has site symmetry 5 and contains tional symbol is needed whenever the c axis of the c tetragonal tetrahedra that have 5 symmetry, $4 complex in standard orientation is to be multiplied by in Schoenflies notation, formerly called spheno- 3 to give the c* length of the cell. PC indicates hedral. The Th atoms in Th3?4 illustrate the S this transformation. P indicates the doubling of F" 1 to mean 444 Ft and similar abbreviations in Schoenflies notation. The Ca atoms in garnet F 1 , J 1 , D 1 , etc,, we can write the following equations: occupy the V* complex (V* = *V + ~V).

15 W Non-intersecting (German: windschief), mutually It may happen that any one of several lattice perpendicular rows of equivalent points. The Cr complexes could be used as generating complex to pro atoms in Cr^Si occupy the W complex. duce, by splitting, a given variant complex. Our first +Y In the invariant complexes +Y + "V = +Y* and task is to set up rules, according to which a unique "y + "Y ( = ~Y*, three equipoints of Y surround choice of generating complex can be made in all cases. one equipoint of Y©, and vice versa, so as to form the letter "Y". The Li atoms in 6.3 Rules for selecting the generating lattice com occupy the "V complex. The Au atoms in Ag3AuTe2 plex (GLCK- occupy Y*. The K atoms in high- temperature The problem may be stated as follows: Given a leucite, K(AlSi 206 ), occupy Y**. variant lattice complex, select a generating lattice As further examples, J* is illustrated by the Pt complex, that is, either an invariant lattice complex atoms in Pt-O^, S* is occupied by Si in garnet (ILC) or a limiting lattice complex (LLC), whose ^, and W* accommodates the B atoms in points will by splitting produce the given variant Zn40(B02 ) 5 . complex. (We note that, if the characteristic space group of the variant lattice complex is different from 6.22 Variant Lattice Complexes.- We have seen that of the generating lattice complex, the generating in Part I that the splitting of the point at the fixed lattice complex can be an ILC or a LLC.) origin of a point group gives rise to a site set, If more than one generating complex is available, which is an assemblage of equivalent points around the consider the site symmetries of the several possible origin. Likewise the points of an invariant lattice generating complexes, taken as positions in the space complex may be made to split into as many assemblages group, and use as many of the following rules as need of equivalent points, the sum of which constitute a ed. variant complex. Examples: Starting from the in variant lattice complex P, in the cubic system (Table Rule 1: If the site symmetries are of different 31), splitting along the principal symmetry directions orders, take the highest order (n in Table B). of the site symmetry gives 6z (see Table 1, last line) and the result is the univariant lattice complex P6z. TABLE B If we take I, instead of P, as the generating complex, the resulting univariant complex will be I6z, which is Possible site symmetries for an invariant lattice complex listed according to decreasing Laue sym composed of two octahedra of equivalent points, one metry for each order n of the point group and each around each of the two points of I. number F of degrees of freedom The direction in which splitting is to take place F=0 F=l F=3 is expressed by using, once, twice, or tnree times, 48 4/m 3 2/m tne letter that designates the variable, as was done for site sets in point groups. Examples: P12xx is a 24 432, ?3m; 2/m 3; 6/m 2/m 2/m univariant complex composed of points that lie on the 16 4/m 2/m 2/m bisectrices of the interaxial angles and thus are the 12 23; 622, 62m; 6/m ; 3 2/m 6mm apices of a cuboctahedron. ISxxx is also univariant, 8 422, 42m; 4/m; 2/m 2/m 2/m 4mm but the length x is plotted three times in succession, 6 B; 32; 3 6; 3m parallel to the three unit vectors a-j, a« and a^, so 4 4; 222; 2/m 4;mm2 that the resultant lies along the body diagonal of the 3 3 cell and, within the cell, the complex consists of two 2 T 2 1 cubes, one around each of the two points of I, the generating complex. In the orthorhombic system (Table 27), F2y2z is made up of four site sets 2y2z, one Example: In space group Pm3h, position 6(0) around each of the four points of the generating com is the characteristic position for a uni van* ant lat plex F; it is a bivan"ant complex, because y and z tice complex. This LC may be described from two dif- f have different independent numerical values. In the ferent generating complexes, both of which are invari same system, a simple trivariant complex is P2x2y2z, ant lattice complexes: P in l(a) with site symmetry the symbol of which is self-explanatory. m$n and s in 3 ^) W1© th S1te symmetry 4/wmm. The

16 order of the group of site symmetry is 48 in l(a) and the y direction, wnich takes precedence over the z 16 in 3(d). P must be chosen as generating complex, direction. and the univariant lattice complex will be written P6z. Example: For Q(d) in 12^2^2^, site symmetry 2 If the site symmetries have the same order, it is available in three orientations; 2.. in 4(a) is may be useful to consider synonyms, that is to say, to cnosen rather than .2. in 4(&) or ..2 in 4(

Rule 4: If the site symmetries are different 6.4 Distribution symmetry.- point groups of the same order and rule 3 does not We have designated (Sn. 1.1) the fixed point of a apply, take the group of proper rotations. point group by the term high symmetry point (HSP). Example: For 16(n) in Ptynbm* site symmetry 422 The generation of a variant lattice complex may b

17 (if no power is stated, all the powers are meant to be operation of the distribution symmetry; the latter used). The symbol of the symmetry element will be must bring 000 onto instead. Likewise in an oriented by means of dots that stand for the several j centered space group, the distribution symmetry columns of the Hermann-Mauguin symbol. It will be should bring 000 onto ^0, not onto 00^. written in front of the symbol of the GLC. Enough Rule 7 is necessary to avoid ambiguity whenever symmetry operations must be stated to specify the Cc is the GLC. Without this rule, two different com orientations of all the subsets. Together they con plexes would get the same symbol, namely: . ,mC 2x2yz stitute the distribution symmetry, which is not a for both Coom(m] and ibam (k) . This situation arises from the fact that C may be constructed as C + 0050 group. 1 c The following rules have been applied in express or as I + OOsI. ing the distribution symmetries. The complete lattice-complex symbol may contain up to four parts: (1) the generating -lattice -complex Rule 1: All the symmetry elements used to ex symbol, which consists of a letter for an invariant press the distribution symmetry should pass between lattice complex (possibly modified by a subscript or the equivalent points of the GLC. a superscript), followed by (2) the site set that indicates the splitting at each point of the GLC, and Rule 2: The number of symmetry elements used preceded by (3) the distribution symmetry. To avoid must be kept to a minimum. ambiguity it may be necessary to specify the crystal system; this is done by prefixing (4) the initial Rule 3: The number of translation components letter of the name of the system, in lower case, and must be kept to a minimum. hyphenating it to the symbol. In addition, in case of a representation symbol, parts (3), (1), (2) may be Rule 4: Symmetry planes are to be preferred to enclosed in a parenthesis modified by a subscript or axes; axes to center; among axes, 4 to 4. a superscript, and the symbol so-far described may be preceded by a shift vector. Rule 5: In order of preference the directions are *, y, z. 7. WEISSENBERG COMPLEXES AS GENERATING COMPLEXES

Rule 6: Whenever possible, splitting is done 7.1 The Weissenberg complexes.- from the standard representation of the GLC, except There exist lattice complexes, 67 in all, for when the chirality of an enantiomorphous GLC is lost which the multiplicity does not decrease for any spec in the resulting variant lattice complex. ial values of the co-ordinates. Such a lattice com Example. In 14,32 the sites of 24(/)2.. can be plex can simulate an invariant lattice complex as a obtained oy splitting from the standard representation limiting complex. These 67 complexes were first re V or from the alternate representation "V. (Either cognized by Weissenberg, who gave them the name Haupt- one will do.) In such a case, instead of using both gitter. We propose to call them the Weissenberg com representations and writing .3. + V"V2z, we omit the plexes. sign and write .3.V2z. (In so doing, we follow a The Weissenberg complexes may be regarded as a rule proposed by Carl Hermann.) In the same space basic set of generating complexes. By splitting, they group, however, we write 4-j.."Y*3xx and 4^.. Y*3xx lead to 402 lattice complexes. The Weissenberg com for the two univariant lattice complexes realized in plexes include all the invariant lattice complexes and 24(0) and 24 (/i), each of which retains the chirality also some univariant, bivariant and trivariant ones. of its GLC: ~Y* and "V, respectively. 1 All of them, except the trivariant ones, could be used as generating complexes; according to our rules, how Rule 7: In a C centered space group, it is not ever, a few of them are not needed. The reason for allowed to bring 000 to coincidence witn 00^ by an this is that we give synonyms of lattice-complex sym bols only when the site symmetry is of the same multi 1 It would be superfluous to repeat the sign in plicity for all the possible generating lattice com front of the univariant complex syiribol and write: plexes; if we were to give all possible synonyms, all

(4 ]L ..~Y*3xx) and . . the Weissenberg complexes would be necessary.

18 7.2 Tables of Weissenberg complexes.- 2,2,..FYlxxx in P2,3. In this©11 example, position (a) The Tables of Weissenberg complexes (Tables 18- of F2-|3 has co-ordinates xxx; [ioi + (xxx)]*. For 24) give the following information, under each crystal special values of the co-ordinates, the position sim system. ulates two invariant complexes, F and Y, each one in Each Weissenberg complex appears as a subheading, four different representations, as follows: for in which its symbol is followed by its characteristic x = 0, F; mfor x = 1/8, *Y; for inx « 1/4, "ill4 F; for space group, data pertaining to the position it oc x = 3/8, iii Y; for x = 1/2, iii F; for x « 5/8, 111 + ^n ^^ cupies in its standard representation (multiplicity, 222 Y; for x = 3/4, ||| F; for x = 7/8, Y. Accord Wyckoff letter and site symmetry) and the co-ordin ing to our rules (Sn. 6.3), we use F as the first gen ates of the equivalent points. erating complex and Y, which stands for the four re Under this subheading comes the list of all the presentations of Y, as the second one. The distri lattice complexes that are derived by splitting. Each bution symmetry 2,2,.. is found by applying the rules one is given in its standard representation only, its (Sn. 6.4). Note that the figure illustrates only the symbol being followed by that of the characteristic standard representations of F and Y; this is done to space group with Wyckoff position letter, and the co avoid cluttering up the drawing. ordinates of the equivalent sites. It may happen that-the co-ordinates of the sites 7.32 Hexagonal and rhombohedral systems, in of the position given in the second column are not hexagonal description.- The z co-ordinates are ex those listed in the third column but are shifted by pressed in twelfths of the c cell edge. a constant vector. 1 In such a case, the Wyckoff There are 7 invariant complexes: P, G, E, N, Q letter is marked by a prime (©). Example: the sixth in the hexagonal system; R and M in the rhombohedral lattice complex listed under P in Table 18 is system (Fig. 54). 4..P23xx, I432(£©); the co-ordinates given in the In Examples of non-shifted representations are il ternational Tables (1952) under -T432 (i) are those of lustrated (Fig. 55). the shifted representation ll| 4..P23xx. There are 5 univariant and 2 bivariant complexes, The co-ordinates are presented in condensed form. in these two systems, that are Weissenberg complexes Moreover, the several origins that are provided by a (Figs. 56-57). centered lattice mode will be indicated by means of In case of a bivariant complex, the range of the conventional lattice letter: I + (xxx; ...) in variation is indicated by hachured triangles. Note stead of (000; 222) + ( xxx *» ) that these triangles do not give the complete 2-dimensional range. The height of each triangle can be 7.3 Figures of Weissenberg complexes.- read at its three corners. Any site inside one of 7,31 Cubic System.- The * ico-ordinates are ex the hachured triangles will have corresponding sites pressed in eighths of the cell edge. in all the others. The 16 invariant lattice complexes are figured in standard orientation (Fig. 49). They are repeated, 7.33 Tetragonal system.- The 2 co-ordinates are with examples of their representations, on Figs.50-51. expressed in eighths of the a cell edge. The equations that relate the complexes and their re The tetragonal invariant lattice complexes are: presentations can be read off these figures. P (alternate representation C = Py), I (alternate re There are five univariant cubic lattice complexes presentation F = I y ), D (which occurs only as VD), T that are Weissenberg complexes (Figs. 52-53). They (which occurs only as VT). Both VD and VT are used as are drawn in standard representation, each in its cnar- generating complexes. Another representation of P is acteristic space group. The full range of values of C . These tetragonal lattice complexes and the alter the parameter that is permitted by the one degree of nate representation F can be found among the figures freedom cannot be shown graphically, but arrows in of the cubic complexes (Figs. 49-51): only the c/a dicate the range between the two or three limiting ratio must be taken t 1. Additional representations cases that are used as generating complexes. Example: C, C , VD, and VT do not appear on the cubic drawings and are presented in Fig. 58i 1 The shift vector is not given in Tables 18-24; There are eight variant tetragonal Weissenberg it is specified in Tables 25-31 and 32-38. complexes: 7 univariant and one bivariant (Figs. 59-

19 60). The symbol is that of the complex in its stand 7.36 Anorthic system.- The anorthic system has ard orientation; the drawing figures the complex in only one Weissenberg complex. It is the anorthic P, the representation realized in the position of the which results from anorthic deformation of any other space group, and the necessary shifts are indicated in P. the legend of the signs used to show the equivalent points of the generating complexes. The shift of the 8. LIST OF THE LATTICE COMPLEXES first generating complex is thus specified and the AND THEIR OCCURRENCES symbol of the figured representation can be obtained 8.1 Arrangement of the Tables.- by prefixing this shift to the given Weissenberg-com- The crystal systems are listed in order of de plex symbol. creasing symmetry (Tables 25-31). The tables can have from 2 to 5 columns. The 7.34 Orthorhombic system.- The z co-ordinates first column lists the lattice complexes, each one are expressed in eighths of the o cell edge. followed by its alternate representations. If needed The orthorhombic invariant lattice complexes are: a footnote ior a column gives the transformation ma P, I, F, D, T, which result from easily visualized de trix 1 from the cell of the standard representation to formations of the cubic lattice complexes that are de the cell of the space group in which the occurrences signated by the same letters (Fig. 49), and C, which of the alternate representation are being considered. results from the orthorhombic deformation of the tet Unit transformations are not explicitly given. ragonal C (Fig. 58), itself a representation of the The next column gives the multiplicity of the tetragonal lattice complex P. lattice complex, in each of its representations, re Alternate representations of orthorhombic in ferred to the cell of the space group. Example: in variant lattice complexes are as follows: P£» the the tetragonal system (Table 28), the multiplicity is orthorhombic analog of the cubic P2 (Fig. 50); Cc , 1 for P, 2 for Po and C, 4 for P . and C . C aD C the orthorhombic analog of the tetragonal GC (Fig.58); The next column gives the site symmetry, ori and additional representations P , P, , P , P, , P , ented in the symbolism of the relevant crystal system. Pab» V ! b© V A © B © V V Bb < R 9© 61) © Example: in the orthorhombic system (Table 29), the There are 5 univariant orthorhombic Weissenberg site symmetry of P . is 222 in Cmma(a) and Cmma(b) , complexes, 2 bivariant arid one trivariant (Figs. 62- 2/m.. in Crrma(c) and Cmma(d) , .2/m. in Crma(e) and 63). Two of them need special explanations. Cmma(f) , ..2/m in Cnrnn(e) and Cmnm(f) , and T in The bi variant complex T.2-, B,A FI Ixz, in Pnma(c)9 Pban(e) , Pban(f) 9 Pmrm(c) and Pmrm(d). i D a a is illustrated in shifted representation by means of The last column lists all the occurrences of two orthogonal projections; the ca projection showing every lattice complex, in its standard and alternate the partial range of site location by means of ha- representations. The first space group mentioned is chured 2-dimensional fields. the characteristic space group and the first Wyckoff The drawing of the trivariant complex, which is letter is that of the position which defines the stand realized in P2^2^2^(a) , shows partial 3-dimensional ard representation. All representations that are range of site location by means of irregular tetra- shifted with respect to the standard one have the hedra. shift specified after the Wyckoff letter by its three fractional components. 7.35 Monoclinic system.- The y co-ordinates are expressed in eighths of the b cell edge. 1 The matrices are presented in the linear form, There are two monoclinic invariant complexes, P long ago proposed by T.V. Barker, in which the three and C in standard representations (Fig. 64). Examples rows are written on one line, separated by oblique of alternate representations, P^, P , P ab and A, I, fraction bars. Example: letting a©, 6©, c© be the C , F, for P and C, respectively, are also illustrated vectors that define the conventional cell of the space because they are used in the text (Tables 23, 30 and group and a, S, c those that define the cell of the 37). lattice complex in standard representation, the trans There are three variant monoclinic Weissenberg formation complexes: two univariant and one bivariant (Fig.65). t© -t, « -*,* - 2t - S will be represented by the matrix 100/010/102. Each crystal system has its table subdivided into The position that serves to define a lattice com four parts, which gather, respectively, the invariant, plex in its characteristic space group is enclosed in univariant, bivariant and trivariant complexes. The a frame. It is the first occurrence of the lattice only exception is the anorthic system, in which only complex in the characteristic space group. Our choice one invariant and one trivariant complex occur. is, in most cases, the same as that made by Carl Her mann. Note that, when the LC occurs in several re 8.2 Remark.- presentations, the standard one is chosen. Note also In the tetragonal system Carl Hermann defined V D that tnere are cases where the LC is defined by means and VT in the smaller cell, whereas we define D and T of a shifted representation. Example: in Pccm, pos in tne larger cell. This difference in outlook ition 8(r)l. This is a consequence of the choice of accounts for the fact that, in Hermann©s tables, the origin. In the example given, if the origin were to transformation matrix for these two complexes is the be chosen at 222 instead of 2/m, the standard re unit matrix, whereas it is yfi/yfi/QN in our own presentation would be obtained. tabulation. In any centrosymmetric space group, the origin is chosen at a center, unless otherwise stated. 9. LIST OF SPACE GROUPS WITH LATTICE-COMPLEX REPRESENTATIONS 9.1 Remarks.- The crystal systems are here listed in order of Even though our method of expressing the orien decreasing symmetry (Tables 32-38). tation of the site symmetry enables us to distinguish Each system is subdivided into as many parts as between 2.. and ..2 in the cubic system, some there are point groups in it. Under each point group, ambiguity persists in each case. Examples: In P^^W, the space groups are taken up in the sequence of in the oriented site-symmetry symbol is 2.. for both creasing superscripts in the Schoenflies symbols. I6z, in 12(fc), and .3.W2x, in 12(i). In 1432, both For every space group, we give, for each position, I12xx, in 24(fc), and ^4..P23xx, in 24(t), have ..2 the lattice complex in the appropriate representation. as site symmetry. The differences between the 2-fold The position data appear under the lattice-complex axes in the two positions in each space group stem symbol. In order to save space, when two consecutive from their relations to the origin. positions representing the same lattice complex have In the hexagonal system, space group £3-jl2, the the same multiplicity and the same oriented site sym co-ordinates of the sites of position 3(a) were given metry, the multiplicity is written in front of the as follows, in the 1935 edition of tne International first Wyckoff letter and the site symmetry, behind the Tables: second. This typographical presentation retains the x, x, 0; x, 2x, 1/3; 2x, x, 2/3. rule, followed by the International Tables, that the In the 1952 edition, the s co-ordinates were increased multiplicity precedes the Wyckoff letter and the site by 1/3, so that the sites are now written: symmetry follows it. The site symmetry is oriented by x, x, 1/3; x, 2x, 2/3; 2x, x, 0; means of dots, which stand for the unused symmetry and agree with the co-ordinates given for position directions of the lattice in the crystal system being 3(a) in space group P3-.21 in the 1935 edition: considered. This scheme is particularly useful to x, 0, 1/3; 0, x, 2/3; x, x, 0. symbolize point groups that belong to lower crystal In the tetragonal system, space group P4/m

21 characteristic space group. position Q(q)m. ., which defines the lattice complex When several lattice complexes can be chosen as ..2 P I2x2y. Note that the [z] operates on the whole generating lattice complexes, we do not explicitly complex, which can be shifted by any arbitrary z value, symbolize those that would be designated by the same and that we have z - 0 in the characteristic space letter. Example: In P4/ncc, 4(c)4.. is described group. It follows that the 2-fold axis (..2) must as olo(..2CIlz) . (The complex has been defined in move with the complex. 1 Pb/nrm, position 2(

22 APPENDIX

SUBGROUP RELATIONS considered. (TABLES 39-43) If we compare corresponding positions in a space The consanguinity between space groups is deter group and in one of its subgroups, we see that one of mined by their subgroup relations. The description three situations may obtain: of the positions by means of lattice complexes pro 1. Tne site symmetry of the position remains un vides a method to follow the effects of a certain sub changed in the subgroup, but the multiplicity is de group relation on all the positions of a space group. creased. A complete discussion of the questions raised by 2. The multiplicity remains unchained, then it subgroup relations for all the space groups would be is the site symmetry that is decreased. an enormous task. We shall, therefore, limit our 3. Both site symmetry and multiplicity can be selves to a presentation of a few examples to show decreased at the same time; this may happen when the what types of relations are to be expected and the subgroup is of index 4 or index greater than 4. kind of results that can be obtained. These three rules apply without any modification In a first example (Fig. AT), we consider the to the description of positions by means of lattice cubic asymmorphic 1 space groups and show subgroup re complexes, whenever the positions have no degree of lations connection P2^3 to its minimal supergroups, freedom. Examples: the relation between I« and ? , the latter to their minimal supergroups, and so on, up which occur as 16(a) in Ia3d and 8(a) in Ia3 respec to the common symmorphic supergroup Im$m. In a second tively, is an example of Rule 1; the complex I, which example (Fig. A2), the filiation goes from Fdad, occurs as 2(a) in Im3m and as 2(a) in Pm3n, illustrates through its minimal supergroups and the supergroups of Rule 2; and the relation between W« in (Pm3n}^(c} and the latter, to the common supergroup Pri3m. In pre V* and S*, which occur in Ia3d as 24(c) and 24 (d) > paring these examples, we were able to take advantage respectively, is explained by Rule 3 (Table 40). of a pre-publication copy of an article on subgroup For positions with degrees of freedom, on the relations by Neubuser and Wondratschek (Priv. Comm., other hand, the naming of the corresponding complex 1958), which was of great help to us. In the figures, depends not only on the site symmetry, but also on a solid line connects a group to a subgroup of index the behavior of the generating (invariant) complex. 2, while a dashed line goes to a subgroup of index 4 Rule 1 can be subdivided into two cases: (Fig. Al) or of index 3 (Fig. A2). Along the left la. The site symmetry of the invariant position margin, the scale indicates the multiples of the low remains unchanged; then the invariant complex splits est order, 12, of group P2,3. A space-group symbol into complexes of lesser multiplicity. Example: enclosed between parentheses and followed by a sub 4.. Y**2xxx in Ia3d and 22.. Y*2xxx in 14^2 (Table script 2 indicates that a block of eight cells, two in 39). each of the three co-ordinate directions, is being Ib. The site symmetry of the invariant position does not remain unchanged; then both the splitting 1 t- space group is called syrmorphic if it contains number and the invariant complex change. Example: a position whose site symmetry is the same as the 4.. Y**2xxx in Ia3a and 4.. I 2Y**lxxx in I$3d (Table point group isomorphic with the space group; hemisyrn- 39). morphia, if it is of the second kind and contains a Rule 2 gives rise to three cases. position whose site symmetry is the highest proper sub 2a. The position and the invariant generating group of the first kind of the isomorphic point group; complex suffer a decrease in site symmetry without any asyrnmorphic, if neither of the above. Examples: Pm3m change in the naming of the generating complex; then is symmorphic, Pn3n is hemisymmorphic, Pm3n is asym the variant lattice complex remains the same. Example: morphic. Site symmetry mZm is available in Pmlm, I6z in Im3m> ]2(e)4m.m, remains I6z in Pm3n, 12(f)rcm2. . 432 in Pn3n, but neither m3m nor 432 is site symmetry (Table 40). in 2b. In the naming of the position, the invariant

23 generating complex remains unchanged, but the point 40. Here a set of points in the supergroup, Ia3d or complex changes; then some distribution symmetry may (Im3m) 2 . is followed in all the subgroups until the have to be specified. Example: I12xx in Im3m(h) and general position 1s reached in a subgroup. In Table ..2 I6z2x in Pm3n(k) (Table 40). 41 the positions in space group Pa3 are related to 2c. The change in site symmetry requires a com those in the minimal supergroups Io3 and Fm3. In pletely different generating complex. Example: Table 42 two space groups, F4^32 and 74^32, have a 5a.. Y**3xx in Ia3d(g) and 22.. P26xyz in Ia3(e) common subgroup P Ja3, Pa3, P2,3 is taken from Fig. and Fddd. Al and the subgroup relations are explained in Table

24 Fig. Al. Subgroup relations in cubic asymmorphic space groups;

432 23

32 (Pm3n)

LIa3a

Fig. A2. Subgroup relations in space groups of different systems,

25 REFERENCES

Belov, N.V., N.N. Neronova, J.D.H. Donnay and G. Hermann, C. (1953) Paper presented at the Deutsche Uonnay (1960) Tables of magnetic space groups. Mineralogische Gesellschaft, Sektion fur Kristall- Acta Cryst. 13 , 1085. kunde, Erlangen meeting. Belov, N.V., N.N. Neronova, J.D.H. Donnay and G. Hermann, C. (1960) Zur Nomenklatur der Gitterkomplexe. Donnay (1962) Tables of magnetic space groups. Z. Krist., US, 142-154. II. Special positions. Jour. Phys. Soc. Japan Internationale Tabellen zur Bestimmung von Kristall 27(B-II), 332-335. strukturen (1935) 2 vols. Berlin, Borntraeger. Boldyrev, A.K. (1925) Die vom Fedorow-Institut ange- International Tables for X-ray Crystallography, vol. 1 nommene kristallographische Nomenklatur. Z. Krist. (1952, 1965, 1969). Birmingham, England, The 62, 145-150. Kynoch Press. Burzlaff, H., W. Fischer and E. Hellner (1968) Die Loeb, A.L. (1967a) Crystal algebra of cubic lattice Gitterkomplexe der Ebenengruppen. Acta Cryst. complexes. Tech. Rept. iso , Ledgemont Labora A24, 57-67. tory, Lexington, Mass. Burzlaff, H., W. Fischer and E. Hellner (1969) Bemer- Loeb, A.L. (1967b) Domains, interstices and lattice kungen zu "Die Gitterkomplexe der Ebenengruppen" complexes: derivatives from the b.c.c. lattice. (Acta Cryst. A24, 57, 1968) und "Kreispackungs- Tech. Rept. 133, Ledgemont Laboratory, Lexington, bedingungen in der Ebene" (Acta Cryst. A24, 67, Mass. 1968). Acta Cryst. A25> 710-711. Loeb, A.L. (1970) A systemative survey of cubic Donnay, J.D.H. (1947a) Hexagonal four-index symbols. crystal structures. J. Solid State Chem. 2, Amer. Mineral. 32, 52-58. 237-267. Donnay, J.D.H. (1947b) The superabundant index in the Niggli, P. (1919) Geometrische Kristallographie des hexagonal Bravais symbol. Amer. Mineral. 32, Diskontinuums. Leipzig, Borntrager. 477-478. Niggli, P. (1920) Lehrbuch der Mineralogie. Berlin, Donnay, J.D.H. (1969) Symbolism of rhombohedral space Borntraeger. (1st ed., pp. 125 et ss.); 2nd ed., groups in Miller axes. Acta Cryst.435, 715-716. 1924, pp. 168 et ss.; 3rd ed., 1941,PP.275 et ss. Donnay, J.D.H., E. Hellner and A. Niggli (1964) Niggli, P. (1928) Kristall ographische und struktur- Coordination polyhedra. Z. Krist, 120* 364-374. theoretische Grundbegriffe. Leipzig, Akad. Donnay, J.D.H., E. Hellner and A. Niggli (1966) Verlagsges. Symbolism for lattice complexes revised by a Kiel Smirnova, N.L. (1966) An analytical method of repre Symposium. Z. Krist. 123, 255-262. sentation of inorganic crystal structures. Acta Fischer, W. (1968) Kreispackungsbedingungen in der Cryst. 21, A33. Ebene. Acta Cryst. A24, 67-81. Smirnova, N.L., and L.I. Poteshnova (1966) Vestnik Hellner, E. (1965) Descriptive symbols for crystal- Moskovskogo Univ., Ser. Geol. No. 6. structure types and homeotypes based on lattice Weber, L. (1922) Das viergliedrige Zonensymbol des complexes. Acta Cryst.jfl, 703-712. hexagonal en Systems. Z. Krist. 57, 200-203. Hermann, C. (1935) Gitterkomplexe. Int. Tab, zur Bestimmung von Kristallstrukturen, vol. i, chapters VI and VII.

26 FIGURES

501-563 0-73-3 Figs. 1-47. Site sets in stereographic projection. Name of site set as co-ordination polyhedron and of its dual as crystal form.

Fig. 1. Point. Monohedron. 2. Line segment (through 0). Pinacoid, 3. Line segment (off 0). Dihedron. 4. Trigon (through 0). Trigonal prism. 5. Trigon (off 0). Trigonal pyramid. 6. Rectangle (through 0). Rhombic prism. 7. Rectangle (off 0). Rhombic pyramid. 8. Tetragon (through 0). Tetragonal prism. 9. Tetragon (off 0). Tetragonal pyramid. 10. Rhombic tetrahedron. Rhombic tetrahedron. 11. Tetragonal tetrahedron. Tetragonal tetrahedron. 12. Regular tetrahedron. Regular tetrahedron. 13. Truncated trigon (through 0). Ditrigonal prism. 14. Truncated trigon (off 0). Ditrigonal pyramid. 15. Hexagon (through 0). Hexagonal prism. 16. Hexagon (off 0). Hexagonal pyramid.

28 8

10 11 12 \

13 U 15 16

29 Figs. 1-47. Site sets in stereographic projection. Name of site set as co-ordination polyhedron and of its dual as crystal form.

Fig. 17. Octahedron. Cube. 18. Trigonal antiprism. Rhombohedron. 19. Trigonal prism & pinacoid. Trigonal dipyramid. 20. Trigonal trapezohedron & pinacoid. Trigonal trapezohedron. 21. Truncated tetragon (through 0). Ditetragonal prism. 22. Truncated tetragon (off 0). Ditetragonal pyramid. 23. Tetragonal tetrahedron & pinacoid. Tetragonal scalenohedron. 24. Tetragonal prism & pinacoid. Tetragonal dipyramid. 25. Tetragonal trapezohedron & pinacoid. Tetragonal trapezohedron. 26. Rectangular parallelepiped. Rhombic dipyramid. 27. Cube. Octahedron. 28. Truncated hexagon (through 0). Dihexagonal prism. 29. Truncated hexagon (off 0). Dihexagonal pyramid. 30. Pinacoid & rhombohedron. Hexagonal scalenohedron. 31. Truncated trigonal prism & pinacoid. Ditrigonal dipyramid. 32. Hexagonal prism & pinacoid. Hexagonal dipyramid.

30 17 18 19 20

21 22 23

25 26 27 28

29 30 31 32

31 Figs. 1-47* Site sets in stereographic projection. Name of site sets as co-ordination polyhedron and of its dual as crystal form.

Fig. 33* Hexagonal trapezohedron & pinacoid. Hexagonal trapezohedron. 34. Truncated tetrahedron. Trigon-tritetrahedron. 35. Cube & 2 tetrahedra. Tetragon-tritetrahedron. 36. Cuboctahedron. Rhomb-dodecahedron. 37. Pentagon-tritetrahedron & 2 tetrahedra. Pentagon-tritetrahedron. 38. Dihexahedron & octahedron. Dihexahedron. 39. Truncated tetragonal prism & pinacoid. Ditetragonal dipyramid. 40. Truncated hexagonal prism & pinacoid. Dihexagonal dipyramid. 41. Truncated cube. Trigon-trioctahedron. 42. Rhombicuboctahedron. Tetragon-trioctahedron. 43. Cube & 2 tetrahedra. Hexatetrahedron. 44. Truncated octahedron. Tetrahexahedron. 45. Cube & octahedron & dihexahedron. Didodecahedron. 46. Snub cube. Pentagon-trioctahedron. 47. Truncated cuboctahedron. Hexaoctahedron.

32 33 35 36

37 38 39 40

41 43 44

45 46 47

33 Order of the group

Fig. 48. The point group at the lower end of a connecting line is a subgroup of that at the upper end. Double (or triple) lines mean that the lower group is subgroup of the upper one in two (or three) non-equivalent orientations. Dashed lines show subgroups of index > 2.

(After C. Hermann, Int. Tab., vol. I, p. 49, 1935.)

34 )- "V. j — — V ) ^ j -«f —

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ru? P-» ^J 0£> Vl- ' F J r -1 - -5- -7- -: - Cj -7- R p ^/ / *- - ^ \- -0- - ' i —H- ^ "~~ *.5- - 1 - -7- --r- 5- - 1 V hS -0- -0- 2 - -0- -0- -2.6 -1 __ tj- -7- -1- -3- -3- -7- _ P> - 1 - 4- J- * < -0-^ • 4» h- -7r h 5 3- 1I - _ F>-h- 1 ^3 r- 1 r _ n.6- d— *1 oc jt-U l VV W* T - 5 —— I-. i • ii i - n 1 1 \ V f-T-37 . i i 15-j-i [-1-15 i | |37-p-, -( > „ _l 1 0 1 1 t, > 1^1 1.0 5 J ^^> 1 B 1 c. 1 V5 — 1 1 -( « i *- — h )H/• 1 > (5 -(5 J -6H > Ci -2- -€ t— «^ -7- 7 5 \- -15--4--37- H -37-H- -1 5- -^ -2- 1 -6- (5-1— i — i-*--,0 f3. 2> 6 -2 o -c)-!-'*- 1-04-44-c)- /I J- -C)-h4 -C H/• >H 1 H- t> \ 1 1 64-^-.->1 j eM 1 o 1 p, 3-1 ———— i— :- 4 1 •2 l_I37 La_l 15 i_l s s* V*

Fig. 49. The 16 cubic invarian t lattice complesxes in standard orientation. Heights in eighths of cell edge.

35 J — ^ u *t

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Q r 6n - /^ ) ———————— v. ) 0 4-dd-04^bjLd04 (D ———— ^*i ———— vr) ^» —————— wr\ ———————yi r I F

c h-*^^ -l c - -3 7- -3 7- c •^•y -1 c * xj *j i xJ i * >J %J / i J V? M .•^ h- -3 7--1 -3 7- i 5- -1 5- -3 7- -1 5- <^ 7- -- -1 5- -3 7- -15- -3 7- -37- -I 5- -3 7- -i 5- -6 -2 > T< h H H < -y T _2 -0 4-L2cJ R 4- -0 4- •0< 2 Ru 0 6 0 L- i- -C)- -^ c)- -( ) 6- -2 6- 2 -0 4- -26- -04- -2^

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^ f5- J- -1 r- i- -1 -5- ^ -4- 4- -2 5- -o »- -26- -0 r - - /^ n i- -1 *- \.5 — i- H- V. V. -t- 6- -04- -2 5- -0 4- -2 H H -1 T- _ c5 — -]L — _^>- -7- ) )4- -26- -04- -26 ^•0 ,**

Fig. 51. Cubic invariant lattice complexes and representations (continued). Heights in eighths of cell edge.

37 Z^./FYlxxx I43d(

Ixxx 2j3.SVlz ® UP OP D Y os ® DOWN

Fig. 52. Cubic univariant Weissenberg complexes in standard representation and characteristic space group. Heights in eighths of cell edge. (See Section 7.31, p. 19.)

38 Ia3faO .3.S*J2 V*lx a s*o J2 O v

Fig. 53. Cubic univariant Weissenberg complexes in standard representation and characteristic space group. Heights in eighths of cell edge. (See Section 7.31, p. 19.) (continued).

39 0//V-0// Fig. 54.

Hexagonal (P, G, E, N, +Q) and rhombohedral (R, M) in variant Weissenberg complexes. Heights in twelfths of c. M

40 Q

-2,«

0,6- 0,6 -0,6 Fig. 55. /7/7 / Examples of non-shifted / / / / representations of the 2,8 hexagonal and rhombohedral Weissenberg complexes. Heights in twelfths of c. ©M,

41 P6,22/W 3,.2 f£ c+Qclx P6,22^/3,2.PCc Ec+Qclxx PCC O +Q

3 ? ..P.+Qlxx c. \f

00 3 PC

Fig. 56. Hexagonal univariant and bivariant Weissenberg complexes. Heights in twelfths of o. (See Section 7.32, p. 19.)

42 32 ..P+Qlxy

3 3C

Fig. 57. One more hexagonal bivariant Weissenberg complex. Heights in twelfths of o. (See Section 7.32, p. 19.)

43

501-563 O - 73 - 4 04- •04

04

04- •04

Fig. 58. Four representations of tetragonal invariant Weissenberg complexes that do not appear on the cubic drawings. Heights in eighths of o.

44 P4/nmm (c) ..ZCIlz 4.. I2 Pc2 lx o o£oc

o-

1(3-——®

•©-

©—-ai

- Pcc oooipcc

Fig. 59. Four tetragonal univariant Weissenberg complexes, Heights in eighths of c. (See Section 7.33.)

45 <§>

P422(tf) 4-.I vDlxx P4,22(c) 31 o c ^ T m JL-i-O vn O J-r *—' A A V V ooo|pcc

(j)@

<2L

®- -(3F

14^^^) 22 OPCC

Fig* 60. Three more univariant tetragonal Weissenberg complexes and the only bivariant one. Heights in eighths of o. (See Section 7.33.)

46 -o o- -0 04- •04

3 C>

3 — C ) ————— 0 ———— C )4 ————————— 0 *J V*) C ) C PO Pb PC

4 0 4 C J*-TM \J04 t VJ*n 4 —————————— 0 3 ————— 0 ————— 0

0 4 04 () 0 0

4 ———— 04 ———— 0 4 C 4 —————————— C 4 C) ————— 0 ————— 0 Pbc PQC Pflb

r D L) C ) ———— 0 ———— 0 0-4. —————————— 04 4 ) C ) 4 4 26 4 j"> r<• ) C) ———— 0 ———— C) 0 * —————————— 04 IQ Ib Ic

0 ———— 4 ———— C) ( ) —————————— (3 C ) —— 4 —— 0 — 4 — 0

i * *

0 ————— 4 ————— (D (D —————————— (3 ( A B A b

0 ———— 4 ———— >«/^ O r\ i3 c ; ——————— b

0 4 0 • 4 4 4

\ Q 0 4 0 0 0 (D AQ B b

Fig. 61. Representations of orthorhombic invariant Weissenberg complexes. Heights in eighths of o. For the invariant complexes themselves, look up cubic P, I, F, D, T (Fig.16) and tetragonal representation C (Fig.58), which are their topological analogs.

47 —— ©(

f©

Pmma (e) .2. PQ Biz Pmmn 2,.. CIlz OC DOO^I

2r .CcFly Cmma 2..Pab Flz ooojc

Immate; .2. Bb Aalz 2.1 PbC Ab CcFixy

Fig. 62. Five univariant and one bivariant orthorhombic Weissenberg complexes. Heights in eighths of c.

48

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of of

orthorhombic orthorhombic

eighths eighths

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One One

63. 63.

Fig. Fig. vo 04-———————————04 0—————————————0 04 ————— 04————04

0 0

04———————————04 0 ———————————— 0 04 ———— 04 ———— 04 Pbh Pc PKab

u «* u / w r*

0i ————— 4 ————0i i i a

Fig. 64. The two monoclinic invariant Weissenberg complexes , in standard and alternate representations. Heights in eighths of b.

50 OOO±PC DO

C 2/c rf?; Cc Fly

A Oiioc

The two univariant and the only bivariant monoclinic Fig. 65. Weissenberg complexes. Heights in eighths of b.

51

TABLES Table 1. Rules of Splitting

Point group: homopolar single, 1st Site 2nd Site 3rd Site Point-complex primary polar Split. Syrnm. Split. Symm Split. Symm. symbol axis axis 1 Ixyz 1 Ixyz I 2xyz 1 Immediate 2xyz ? 4xyz 1 splitting 4xyz 3 6xyz 1 6xyz * m Ixz m 2y 1 Ixz2y 2/m 2xz m 2y 1 2xz2y 4/m 4xy m. . 2z 1 4xy2z 3/m ** 3xy m. . 2z 1 3xy2z 6/m 6xy m. . 2z 1 Stay in 777> 6xy2z 4 m 2 4xz .m. 2y 1 then out. 4xz2y 42m 4xxz . .m 2y 1 4xxz2y 3ml 6xxz .m. 2y 1 6xxz2y 3 1 m 6xz . .m 2y 1 6xz2y 3 m 6xxz ,m 2y 1 6xxz2y 2 ly 2 2xz 1 Iy2xz 4 lz 4.. 4xy 1 Iz4xy ,3 1 1 Iz 3.. 3xy 1 Iz3xy •3 1 Ixxx 3. 3yz 1 Ixxx3yz 6 lz 6.. 6xy 1 Iz6xy 222 2x 2.. 2yz 1 Stay on axis, 2x2yz 422 4x .2. 2yz 1 4x2yz (3 2 1 3x .2. 2yz 1 then off. 3x2yz ]3 1 2 3 xx ..2 2yz 1 3xx2yz T *** l3 2 3xx .2 2yz 1 3xx2yz' 622 6x .2. 2yz 1 6x2yz 2 3 6z 2.. 2xy 1 6z2xy 4 3 2 6z 4.. 4xy 1 6z4xy 2 m m Ix 2mm 2y . .m 22 1 Ix2y2z m 2 m ly m2m 2z m. . 2x 1 Iy2z2x m m 2 lz mm2 2x .m. 2y 1 Iz2x2y ! 4mm lz 4wn 4x .m. 2y 1 Iz4x2y (3 m 1 lz 3m. 3xx .m. 2y 1 Iz3xx2y <3 1 m lz 3.m 3x . .m 2y 1 Iz3x2y (3 m Ixxx 3m 3y .m 2z 1 Ixxx3y2z 6mm lz 6mm 6x . .m 2y 1 Iz6x2y m in m 2x 2mm 2y . .m 2z 1 2x2y2z 4/m m m 4x m2m. 2y m. . 2z 1 4x2y2z 3/m 2 m 3x m2m 2y m. . 2z 1 3x2y2z 3/m m 2 3 xx mm2 2y m. . 2z 1 3xx2y2z 6/m m m 6x m2m 2y m. . 2z 1 6x2y2z 7717} 3 6z 2mm. . . 2x m. . 2y 1 6z2x2y 43m 6z 2. mm 2xx . .m 2y 1 6z2xx2y m 3 m 62 4m.m 4x m. . 2y 1 6z4x2y Stay on axis; off axis, in m; out of m. The symbol is the same in hexagonal and rhombohedral axes. ** m m 6 *** The same symbol serves for 312 in the hexagonal system and for 32 in rhombohedral co-ordinates. The preceding symbol is used both for 321 in the hexagonal system and for 32 in hexagonal co-ordinates.

54 Table 2. Site Sets 1 Found in the Cubic System

Co-ordination polyhedron. Dual crystal form. Site-set symbol . Co-ordinates. Occurrences.

Tetrahedron . Tetrahedron.

4xxx: xxx; xxx K 43m(a); 43m(b); 23(a); 23(b). Octahedron. Cube.

6z: ±(00z) K m3m(a); 432(a); m3(a); 23(c). Cube. Octahedron.

8xxx: ±(xxx; XXXK) m3m(b); 432(b); m3(b). Cuboctahedron. Rhomb-dodecahedron. 12xx: ±(x x,0) m3m(c); 432(c). Dihexahedron + octahedron. Pi hexahedron (or pentagon-dodecahedron).

6z2x: ±(x,0,±z) < m3(c) Truncated tetrahedron. Tri gon-tri tetrahedron. (For Cube + 2 tetrahedra. Tetragon-tritetrahedron. (For 6z2xx: (xxz; xxz)< ; xxz TT 43m(d). Pentagon-Tritetrahedron + 2 tetrahedra. Pentagon-Tri tetrahedron. 6z2xy: (xyz; xyz; xyz; xyz)< 23(d). Truncated octahedron. Tetrahexahedron. 6z4x: ±(x,0,±z) TT m3m(d). Truncated cube. Tri gon- tri octahedron. (For Rhombi cuboctahedron. Tetragon-tri octahedron. For 6z4xx: ±(x,x,iz K; xxz TT) m3m(e). Snub cube. Pentagon-trioctahedron.

6z4xy: (xyz; xyz; xyz; xyz; yxz; yxz; yxz; yxz)< 432(d). Cube + octahedron + dihexahedron. Pi dodecahedron. 6z2x2y: ±(x,y,±z; x,y,±z) K m3(d). Cube + 2 tetrahedra. Hexatetrahedron. 6z2xx2y: (xyz; xyz; xyz; xyz)iT 43m(e). Truncated cuboctahedron. Hexoctahedron. 6z4x2y: ±(x,y,±z; x,y,±z) TT m3m(f).

Limiting cases not included.

55 Table 3. Site Sets 1 Found'in the Hexagonal and Rhombohedral Systems (in hexagonal co-ordinates)

Co-ordination polyhedron. Dual crystal form. Site-set symbol. Co-ordinates. Occurrences.

Point. Monohedron. Iz: OOz 6nm(a); 6mm(b); 6(a); 6(b); 3ml(a); 3ml(b); 31m(a); 31m(b); 311(a); 311(b). Line segment (collinear with origin). Parallelehedron. 2z: 0,0,±z 6/mmm(a); 6m2(a); 62m(a); 622(a); 6/mCa); 6(a); 5ml(a); 31m(a); 321 (a) 312(a); 3ll(a). Trigon coplanar with origin. Trigonal prism. 3xx: xxO 5m2(b); 5m2(c). 3x: xOO 1 S2m(b); 321(c); 62m(c); 321(b). Trigon. Trigonal pyramid. Iz3xx: xxz 3ml(c). Iz3x xOz 1 31m(c). Iz3xy: xyzj 311(c). Truncated trigon coplanar with origin. Ditrigonal prism. 3xx2y: (xyO, yxO) Sm2(e). 3x2y: (xyO ) l 52m(e). Truncated trigon. Ditrigonal pyramid.

Iz3xx2y: (xyz, yxz) 3ml(d). Iz3x2y: (xyO

Hexagon coplanar with origin. Hexagonal prism.

6x: (±x,0,0) 6/mmm(b); 622(b); 3ml(b). 6xx: ±(xxO ) l 6/mmm(c); 622(c); 3lm(b). 6xy: ±(xyOj) 6/m(b). Hexagon. Hexagonal pyramid. Iz6xx: (xxzT ) l 6mm(c), Iz6x: (±x,0 f z) l 6mm(d). Iz6xy: (xyz, xyz) l 6(c). Trigonal prism + pinacoid. Trigonal dipyramid. 3xx2z: (x,x,±z) 5m2(d). 3x2z: (x,0,±z) T 62m(d). 3xy2z: (x,y,±z) T

Limiting cases not included.

56 Table 3 (concluded).

Trigonal trapezohedron + pinacoid. Trigonal trapezohedron. 3x2yz: (xyz; yxz) T 321(d). 3xx2yz: (xyz; yxz) 312(d).

Trigonal antiprism. Rhombohedron. oxxz: ±(xxz) 3ml(c), 6xz: ±(xOz) T 3lm(c). 6xyz: ±(xyz) T 311(b).

Truncated hexagon coplanar with origin. Dihexagonal prism.

6x2y: ±(xyO ) 6/mmm(f).

Truncated hexagon. Dihexagonal pyramid.

Iz6x2y: ((xyz; xyz)^ 6mm(e).

Pinacoid + rhombohedron. Hexagonal sealenohedron. 6xxz2y: ±(xyz; yxz) 3ml(d), 6xz2y: ±(xyz ) T 3lm(d).

Truncated trigonal prism + pinacoid. Di trigonal d^ipyramid. 3xx2y2z: (x,y,±z; y,x,±z) 5m2(f). 3x2y2z: (X,Z,±Z T) I T 6Zm(f). Hexagonal prism + pinacoid. Hexagonal di pyrami d• 6x2z: (±x20,±z) 6/mmm(d). 6xx2z: (X,X,±Z T) T 6/mmn(e). 6xy2z: ±(x,y,±z) 1 6/m(c).

Hexagonal trapezohedron + pinacoid. Hexagonal trapezohedron. 6x2yz: (xyz,xyz,yxz,yxz) T 622(d).

Truncated hexagonal prism + pinacoid. Di hexagonal di pyrami d. 6x2y2z: ±(x,y,±z T ) i 6/mmm(g).

57 Table 4. Site Sets 1 Found in the Rhombohedral System (in rhombohedral co-ordinates)

Co-ordination polyhedron. Dual crystal form. Site-set symbol. Co-ordinates. Occurrences.

Point, flonohedron (pedion).

Ixxx: xxx 3m(a); 3m(b); 31(a); 31(b).

Line segment (collinear with origin). Parallelehedron (pinacoid) . 2xxx: ±(xxx) 3m(a); 32(a); 3l(a).

Trigon coplanar with origin. Trigonal prism. 3xx: xxO K 32(b); 32(c).

Trigon. Trigonal pyramid.

Ixxx3z: xxz < 3m(c). Ixxx3yz: xyz K 31(c) .

Hexagon coplanar with origin. Hexagonal prism.

6xx: ±(xxO K) 3m(b).

Truncated trigon. Pi tri gonal pyrami d . Ixxx3y2z: xyz TT 3m(d).

Trigonal trapezohedron + pinacoid. Tri gonal trapezohedron . 3xx2yz: (xyz, xzy)< 32(d).

Trigonal anti prism. Rhombohedron. 6xxz: ±(xxz K) 3m(c). 6xyz: ±(xyz <)

Pinacoid + rhombohedron. Hexagonal scalenohedron.

6xxz2y: ±(xyz w) 3m(d).

1 Limiting cases not included.

58 Table 5. Site Sets 1 in the Tetragonal System

Co-ordination polyhedron. Dual crystal form. Site-set symbol. Co-ordinates. Occurrences

Point. Monohedron (pedion). lz: OOz 4mm(a); 4mm(b); 4(a); 4(b). Line segment ( coll i near with origin). Parallelehedron (pinacoid) . 2z: ±(00z) 4/mmm(a); 42m(a); 4m2(a); 422(a); 4/m(a); Tetragon (square) coplanar with origin. Tetragonal prism. 4x: ±(xOO) T 4/mmm(b); 42m(b); 422(b). 4xx: ±(x,±x,Q) 4/mmm(c); 4m2(b); 422(c). 4xy: ±(xyO,yxO) 4/m(b). Tetragon. Tetragonal pyramid. Iz4x: ±x,0,z; 0,±x,z. 4mm(c). Iz4xx: +x,±x,z. 4mm(d). Iz4xy: (+++, — +) (xyz, yxz) . 4(c). Tetragonal tetrahedron. Tetragonal tetrahedron. 4xz: ±x,0,z; 0,±x,z. 4m2(c). 4xxz: xxz; xxz; XXZT _ _ 42m(c). 4xyz: xyz, xyz, yxz, yxz Truncated tetragon coplanar with origin. Pi tetragonal prism.

4x2y: ±(xyO; yxO) T 4/mmm(f). Tetragonal tetrahedron + pinacoid. Tetragonal scalenohedron. 4xz2y: ±x,±y,z; ty.tx.z^ _ 4m2(d). 4xxz2y: (xyz; xyz; yxz; yxz) T 42m(d). Truncated tetragon. Pi tetragonal pyramid . Iz4x2y: ±x,±y,z; ±y,±x,z. 4mm(e). Tetragonal prism + pinacoid. Tetragonal di pyramid. 4x2z: ±(x,0,±z) T _ 4/irmm(d). 4xx2z: ±(x,x,±z; x,x,±z) 4/mmm(e). 4xy2z: ±(x,y,±z; y,x,±z) 4/m(c). Tetragonal trapezohedron + pinacoid. Tetragonal trapezohedron. 4x2yz: xyz; xyz; xyz; xyz; yxz; yxz; yxz; yxz 422(d). Truncated tetragonal prism + pinacoid. Di tetragonal di pyramid. 4x2y2z: (x,y, z; y,x,±z) T 4/mmm(g).

Limiting cases not included.

59

501-5SS 0-73-5 Table 6. Site Sets 1 Found in the Orthorhombic System

Co-ordination polyhedron. Dual crystal form. Site-set symbol. Co-ordinates. Occurrences.

Point. Monohedron (pedion). Iz: OOz. mm2(a), mm2(b)

Line segment (col linear with origin). Parallelehedron (pinacoid). 2x: ±xOO. mmm(a); 222(a). 2y: ±OyO. mmm(b); 222(b). 2z: ±00z. mmm(c); 222(c).

Line segment. Dihedron,

Iz2y: 0, ±y f z mm2(c). Iz2x: ±x, 0, z mm2(d).

Rectangle coplanar with origin. Rhombic prism. 2y2z: ±(0, y, ±z) mmm(d). 2x2z: ±(±x, 0, z) mmm(e). 2x2y: ±(x, ±y, 0) mmm(f).

Rectangle. Rhombic pyramid.

Iz2x2y: ±x, y, z; ±x, y, z mm2(e)

Rhombic tetrahedron. Rhombic tetrahedron. 2x2yz: xyz; xyz; xyz; xyz 222(d).

Rectangular parallelepiped. Rhombic dipyramid. 2x2y2z: ±(x, y, ±z; x, y, ±z) mmm(g).

1 Limiting cases not included.

60 Table 7. Site Sets 1 Found in the Monoclinic System

Co-ordination polyhedron. Dual crystal form. Site-set symbol . Co-ordinates. Occurrences.

Point. Monohedron (pedion). ly: OyO 2(a), 2(b). Ixz: xOz m(a).

Line segment (collinear with origin). Parallelehedron (pj_n_acoi_d) . 2y: ±(OyO) 2/m(a). 2xz: ±(xOz) 2/m(b).

Line segment. Dihedron. Iy2xz: xyz, xyz 2(c). Ixz2y: xyz, xyz m(b).

Rectangle coplanar with origin. Rhombic prism. 2xz2y: ±(x, ±y, z) 2/m(c).

Table 8. Site Sets 1 Found in the Anorthic (Triclinic) System

Co-ordination polyhedron. Dual crystal form. Site-set symbol. Co-ordinates. Occurrences,

Point. Monohedron (pedion).

Ixyz: xyz l(a).

Line segment (collinear with origin). Parallelehedron (pinacoid). 2xyz: ±(xyz) T(a).

1 Limiting cases not included.

61 .

l

.m

..2

.

C4xx)

(4xx)

m.

4xx

4xx m.m2

.

m

.

mm

.

l

:OOs: 2.

2mm.

4.

2.

6z

6z 4m.

6z

.m.

(4x)

.m. (4x)

.2.

(4x) (4x) (4x)

m.

4x 4x

m2m.

4x

.

.mm

;OOs.*

2mm.

Iz 2z Iz 4.. 4mm 2z 2..

2 4mm

4.. 2z

2z

2z 2z 4.

.3. .3.

.3.

,3m

.3m

:xxx:

8xxx

4xxx 8xxx

4xxx

1

.m

,m

.

(4xxz)

(Iz4xx) (4xx2z)

;xxz:

(4xx2z)

Iz4xx

4xx2z

4xxz

symbol.)

1

.

.m.

(Iz4x) (4x2z)

(4x2z) (4x2z)

(4xz) :xQz:

Iz4x

System

4xz

4x2z

.m

.

..2

(12xx)

(12xx) (12xx)

:xxQ:

12xx

m.

m.m2

.

1

(4x2y)

(4x2y) (4x2y) (4xy)

(4xy)

(4x2y)

:xy'd;

4xy

4x2y

m.

m.,

.

site-symnetry

Tetragonal

.m

1

Groups

.

1

ill

1

ill ill

1

ill

(6z4xx)

:xxz: (6z2xx)

(6z4xx)

6z2xx

6z4xx

:xyz:

12.

Iz4xy Iz4x2y

4x2yz

4xxz2y 4x2y2z

4xz2y

4xy2z

4xyz

Point

oriented

Table

in

4^

422 422

4

m

(4m2

the (42/77

.

ill 1

1

by

m.

(6z4x)

:xQz:

(6z2x) (6z4x)

6z4x

6z2x

Positions

System.

.

in

.m

l

.

.2.

(6x)

:xQQ:

(6x)

(3x)

3x

6x 6x

m2m

m. m2m

m.

m*

accompanied

i l

l

Sets

:xyz:

Cubic

6z4x2y

6z2x2y 6z2xx2y

6z2xy 6z4xy

is

.

l

9.

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:xxQ:

(6xx)

(3xx)

(6xx) (6xx)

(6xx)

3xx 6xx

m. 6xx

mm2

Site

m.

mm2

symbol

m

Table

Urn

432

23

m

.

m

•OOs-

3.m

3m.

Iz

3.. 2z

2z 6mm

2z

Iz 2z

2z

6mm

6.

6..

site-set

l

.m.

:xxz: (6xx2z)

(6xx2z)

(Iz6xx) (6xx2z)

(3xx2z)

Iz6xx

3xx2z 6xx2z

(Each

,m

.m

1

1 1

1 l

.

(6x2z)

3x2z

(6x2z)

(3x2z)

(Iz6x) (6x2z) Iz6x System

6x2z

.

l

m.

m..

m.

:xyO:

(6x2y)

(6xy)

6x2y

3x2y

3xx2y

6xy

3xy

Hexagonal

1

l

1 1

1

l 1

l

10.

z6x2y

:xyz:

1

3xx2y2z

3x2y2z

6x2y2z Iz6xy

6x2yz

6xv2z

3xy3z

Table

o2w

622

6m2

6mm

622

nvwn

6

m

to

en .

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;00a: (2z) (2z)

2z Iz

mm2 mm2

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.

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.

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(2y)

;0i/0:

2y ly m.

m2m

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(2x) (2x)

2mm 2mm

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2..

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1

m 1

.m .

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.

(2x2y)

2x2y Ix2y

Iy2x

1 1

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i

2

m. System

.m. .m.

.

(2x2z)

Ix2z (2y)

2x2z Iz2x

2y ly

System

•

z

.

i

m m

tjc\j

(2y2z)

Iz2y

•'

2y2z Iy2z

Ixz

(2xz)

m. m.. m..

2xz

•

Orthorhombic

Monoclinic

Anorthic

z

i 1

i i

1 1

1 l

i

4-Czy

Iz2x2y

.* Ix2y2z

2x2y2z

Iy2z2x

2x2yz

2xz2y Ixz2y

Iy2xz

Ixyz

2xyz

13.

14.

15.

---

2mi

Table WI»«Q

222

£

W?n

2

rnhn

1

Table T m

m

<

)

\ /

1

i

».

.m

r

hy UA

.2.

.

..m

.2. (6x) f

i v (3x)

(6x)

(6x) (3x)

6x

3x

1

*

~

o

1

1 x

11

.2

^yy

^

.m.

.

.m.

/

(6xx) ..2

i \jAAy

(6xx)

(6xx) (6xx)

(3xx)

6xx

3xx

are

3ml,

the

•

Their

1

are sets

X3C\)

2/ml,

3m.

.2

2z Iz

3.m

Iz 3.m 3m. .2

2z

«*

Iz (6xx)

2z 2z (6xx)

2z

(3xx)

6xx

3xx

T

site

for

*

1

13..

13.. 13..

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.m.

(6x2z)

6xxz

(3x2xz)

Iz3xx Iz6xx (6xxz)

(Iz3xx)

1 11

11

10

£3CZ

,m

.m

.*

(6xxz)

Ixxx3z (6xxz) (Ixxx3z)

6xxz

co-ordinates. co-ordinates

System

1

Table

.m

(6xx2z)

.

(Iz6x)

6xz (3x2z) (3xx2xz)(3xx2z)

Iz3x (6xz)

(Iz3x)

•

rhombohedral

in

X322C

.*

3m

2xxx Ixxx 3m

2xxx Ixxx

2xxx

the

hexagonal

1

1 1

1

1 1

11

given

(6xx2y)

(6x2y)

(3xx2y)

(3x2y)

(6xy) (3xy)

(3xx2y)

rhombohedral

in

•

Rhombohedral

Z

in

311.

1

13.

Xlj

Table those

1

1 11113.. .*

1 1

1

Ixxx3y2z

6xxz2y

3xx2yz

Ixxx3yz

6xyz

11.

6xxz2y

6xz2y Iz3xx2y Iz3x2y

3x2yz 3xx2yz

Iz3xy

oxyz

In

as

311,

1

3,2

32

Table 4 3m

31 described 31 same descriptions

321,

(311

[M [312

[sim

(321 Iz Iz

2z 2z 002 2z 2z 2z

2z

:00z:

Oz/0 6z 6z 6z

6z 6z

:00z:

6/mmm)

4x

4x 4x

:xOO:

xOO

:

(except

xxx

:xxx:

:

8xxx 8xxx 8xxx

4xxx Ixxx

4xxx 2xxx Ixxx

2xxx 2xxx

:

groups

3xx

xxO

xxO 6x,x

:

xx

point

:xxO:

12xx

12

4xx 4xx

4xx

xxQ

xxz

holohedral

of

:xxz:

Iz4xx 4xx2z

6z4xx 6z2xx

4xxz

xxz

:xxz:

Ixxx3z

6

Degradation

Qyz

subgroups

Cubic

:xOz:

Iz4x

4x2z 4xz

xOz 16.

:zOx:

6z4x

6z2x

oriented

in

Table

xyO

:xyO:

4x2y

4xy

xyz:

ocyz

xyz

:xyz:

:

:xyz:

6z4x2y 6z4xy Iz4x2y

6z2x2y 6z2xy Iz4xy 4x2y2z 4xz2y Ixxx3z2y 6z2xx2y

4xxz2y

4x2yz 4xy2z 4xyz 3xx2yz

6xxz2y Ixxx3yz

6

a

representations

m

their

o

and

r

3m

32 3 3

sets

t

4

42m

4m2 m

4mm

4 4

—mm m

^

'422

•

Site

2

m m

. '

.

c

3m

3m rc.m

23 . 43m 432

m3m 4 • m3 4m. ,32 IT! 42. 4m ,3. 42.2 M- .3.

4., 4.

—

m* zAxT T T T T T T zAxs T T III I : zAx : zA^xxT ., '' c" *" XT ZA^XT AT zxsAT 2 -2- ©Z-

ZT /xjzj o o u ZXXT AZZXXT Ul' HI * • Ul* * zAT XSZAT • -ui

zxT ASZXT Ul 'Ul' 'UI*

AXT zsAxT UI * * * *UI * "Ul ASZXXS .* xszAs " ui .111^ t m t ASZXS Axs 2 ui^ § .^ui . .ui z, zsAxs ?" "s" "s :oAo: :ZQX: :zAx:

17^ G7 vv*^*^G 7 W***^ G7 *7*j /\^X^7yy A.A. ^7 o C o 0 O G

"7 7 ^7 V 7 *7 X~7V 7 XXT ZSXXT ASxxT zsAsxxT uis *ui jui'ui ZT xx 2zT xx 2 zT A2xx2zT uiui • s uiui • s XT z 2xT ASxT ZSA^XT uiuis AT Z^AX X^AI z^xsAT uiSUi 'uisui ZT Agzt x^zj ASXSZT Suiui 'uiuis * * suuu zs xxs XX S z 2xx 2 Z 2xx 2 A2xx 2 2SA2XX3 UIUI *U1 UlUl'Ul

zs AS xs z 2A2 Z 2x2 A2x3 ZSA2X S UIUIUI * UIUIUI • * UIUIUI : ZQO : '• oAo : : 00X * : zAo i : ZQX : : 0Ax : :zAx : x

3x

6 6x

#00 6x

:xOO:

6/mmm

of

xx

3xx

6 6xx

j^cO

6xx

:xxO:

3xx

description)

subgroups

z

Iz Iz 2z Iz 2z 2z 2

Iz

2z Iz OOs 2z 2z 2z 2z 2z 2z

:00z:

:

oriented

(hexagonal

xxz

in

xxz

:

Iz6xx 3xx2z

Iz3xx

6xx2z

6xxz

:

xOz

Degradation xOz

:

3x2z Iz6x

6x2z Iz3x

6xz

representations

:

xyO

Hexagonal

:

3xx2y 3x2y

3xy 6x2y

6xy

their

,

17

and

Table sets

xyz

:xyz:

Iz6x2y

3xx2y2z 3x2y2z Iz6xy

3xy2z 6x2y2z Iz3xx2y

Iz3x2y 6x2yz 6xy2z 3x2yz Iz3xy

6xxz2y 3xx2yz

6xz2y

6xyz

Site

m

h

m

6m2 m 62m

6mm 3ml 622 321 6 6 31m ill m 3

312 3 o 0 o XXX O X X X CN rH CM X CM rH

IX IX 1 X 1 x X XX CM rH CM X CN rH

0 N N N 0 o CN rH CM o CM N

N 1 X N Csl CM X CM Is) N tSl 1 X 1 X CM IX '* X X N X CM rH rH CM rH

N N X IS] 0 CM CM CM O ISJ N X X N X X X X CM rH rH CM rH

>> o >> >^ CM o CM CM 1 X X XX CM rH rH CM rH

N N >> N CM CM CM CM >> NCM |>,NCN >-,t>*»N >-> >> X >> CN CM NCMCN NCN X>,CN N CM CM CN 1 X >, IX N >> IX N CM CN 1 X X N X X XXXXXXlslXX 1 CM rH rH rH CNCNCMrHrHrHrHrHrH 1 CM rH

CN| E

CM| g E E CM CM CMJ E • • E • -CM E CM E CN CM | E • E • -CM CM| £ CM E £ CN CM 1 E E • • CM IrH rH

67

Weissenberg Complexes

Table 18: Cubic system

P Pm3m Ka)m3m 000

P4XXX P43m(e) xxx;xxx< P6z Pm3m(e) ±(00z)< P8xxx Pm3m(g) ±(XXX;XXXK) P12xx Pm3m(i) ± (X3 ±X> 0) K 22. .P 2xxx Ia3(c) J± (xxx); { \0\I± (xxx) }K 4..P2 3xx 1432 (i f ) { I-h (xxO) ;\0\I+ (xxO; xOx; Oxx) } K 4..P2 6xx Im3m(i r ) { J± (xxO) ; \0\ I± (xxO;xOx; Oxx) } K

P6z2xx P43m(i) (xxz ; xxz ) <; xxz TT P6z2x Pm3(j) ±(X,OJ±Z)K P6z4x Pm3m(k) ±(XjOs±z)T! P6zM-xx Pm3m(m) ±(XjXj ±ZK;XXZT\) ..2P 2 6z2x Fm3e(i) {F±(x,0,±z);H&±(±*,0,x)}<

P6z2xy P23(j) (xyz;xyz;xyz;xyz) < P6z2xx2y P43m(j) (xyz;xyz;xyz;xyz)i\ P6zUxy P432(k) (xyz;xyz;xyz; xyz; yxz; yxz; yxz; yxz) K P6z2x2y Pm3(l) + (x lj +Z ' X V +Z ) K P6z4x2y Pm3m(n) + (x v +z*x u +2^11 22..P 6xyz Ia3(e) {I±(xyz); \0\I±(xyz;yzx;zxy)}K . .nP 6z2xy F43c(h) { F+ (xyz;xyz;xyz; xyz); \ \ ^F+ (yxz; yxz; yxz; yx m. . P 6z4xy Fm3c(j') {F+ (xyz; xyz; xyz; xyz; yxz; yxz; yxz; yxz); {7.2P2 6z2x2y} l^F+(xyz;xyz;xyz;xyz;yxz;yxz;yxz;yxz)}K

I Im3m 2(a)m3m 000;\\\ I

IM-xxx I43m(c) I+(XXX;XXXK) I6z Im3m(e) I±(00z)< 1 8 xxx Im3m(f) I±(xxx;xxx<)

1 12 xx Im3m(h) I± (X; ±X; 0) K . . 2I4xxx Pn3m(e) xxx; XXXK; 222+ (xxx; XXXK)

I6z2xx I43m(g) J+ { (xxz; xxz )<; xxz IT } I6z2x Im3 (g) J±rXj^±2jK I6z4x Im3m(j) I±(x3 0,±z)i\ I6z4xx Im3m(k) I± (x3 Xj±z K; xxz TT ) ..2I6z2x Pm3n(k) ±{Xj 03 ±z; m+(±ZjOjX) }< . . 2I6z2xx Pn3m(k) {xxz; xxz; xxz; xxz; \\\+ (xxz; xxz; xxz; xxz )}<

I6z2xy I23(f) I^(xyz;xyz;xyz;xyz)< I6z2xx2y I%3m(h) I+(xyz;xyz;xyz;xyz)-n I6z4-xy I432(j) J+ (xyz; xyz; xyz; xyz; yxz; yxz; yxz; yxz ) K I6z2x2y Im3(h) J± (xy y^ ±z; Xj y^±z)< I6zUx2y Im3m(l) I±(x y^±z;x^y^±z)^ . .cI6z2xy P43n(i) {xyz; xyz; xyz; xyz; H i+ (yxz; yxz; yxz; yxz ) } < ,.2I6z2xy P4 932(m) {xyz;xyz;xyz;xyz;^^(yxz;yxz;yxz;yxz)}< n. .I6z2xy Pr&(h) {xyz;xyz;xyz;xyz;l^+(xyz;xyz;xyz;xyz)}< ru . IBzM-xy Pn3n(i) {xyz; xyz; xyz; xyz; yxz; yxz; yxz; yxz;

69 . ,2I6z2x2y Pm3n(l) ..2I6z2xx2y PnZm(l) }TT d.2I 6z2xy Fd3o (h) F+ {xyz; xyz; xyz; xyz; i J 4+ (yxz; yxz; yxz; yxz);

Pm3m 3(c)4/mm.m

.3.J2x Pm3(f) . 3.J4x PmZm(h)

Fm3m 4(a)m3m

F4xxx F43m(e) F+(xxx;xxx<) F6z Fm3m(e) F±(00z)\c F8xxx Fm3m(f) F±(xxx;xxx<) F12xx Fm3m(h) F± (x, ±x> 0) K bc..F2xxx 4 .F3xx P4232(k') [xxO; \Q\+ (xxO; xOx; Oxx) } K 5.. F6xx Pntmtt 1 ) ±{xxO; JO J+ (xxO; xOx; Oxx) } < 52..F 3xx Fd3c(g') F+{xxO; Ii0l+(xx0;x0 J 4 4+ TarcO; xto; 05x; } K

F6z2xx F43m(h) ; xxz ) <; xxz i\ } F6z2x F6zHx Fm3m(j) FGzHxx Fm3m(k) F± (x9 Xj ±z K; xxz TT )

F6z2xy F+(3cyz;xyz;xyz;xyz)< F6z2xx2y F43m(i) F+ (xyz; xyz; xyz; xyz ) TT F6z4xy F432(j) F+(xyz;xyz;xyz;xyz;yxz;yxz;yxz;yxz)< F6z2x2y Fm3(i) Fm3m(l) bc..F6xyz ± {xyz; 404+ (xyz; zxy; yzx) } K

4(a).32 .. Y3xx ; xOx; {lll+(xyz;yxz);lli+(5%z;zxy;yzx;yxz;zyx;xzy)}<

W Pm3n

,3.W2z Pm3n(g) {.3.J*2z} Pm3n(j)

Im3m 6(b)4/mm.m

.3.J*2x ImM(g)

70 Ed 3m 8 (a) 4 3m

D6z Fd3m(f) ..2D4xxx Fd3m(e) F+{xxx; XXXK; iiU+ (xxx; XXXK) }

..2D6z2xx Fd3m(g) F+ {xxz; xxz; xxz; xxz; i i IT }K

..2D6z2xy F4 32(h) F+{xyz;xyz;xyz;xy_z; JJi }K d..D6z2xy FdZ(g) F-^-{xyz;xyz;xyzjxyz; JJ j ..2D6z2xx2y Fd3m(i) F+{xyz;xyz;xyz;xyz; J j ;a^2j;

+Y- 14 32

22..Y*2xxx 14 32 (e) 4 3 .. tY^3xx (xxO) ; 1 ; Oxx)

22..Y^3xx2yz 14 2 (i) w* Im3m 12(d)4m. 2 I±(OW<

I43d

. 3 . S2z I43d(d) l z) K

. 3dS4xyz I43d(e) ll+ (yxz; yxz; xyz; xyz)

+V 14^^32 isr^;s. 22 I+(OUS OW«

.3.V2z I4 1 22(f)

T Fd3m W^.Sn F+ riii;iiJ K ;

22. .T3xx F4 32(g) Fdim(h) ; I i i ± f£50; 5:05; <9x^j } K

Ia3d

4..Y**2xxx Ia3d(e) . ,I 22xxx} Ia3d(g) J± { H s ; xOx; Oxx) } K

..Y**3xx2yz Ia3d(h) I± { I H ; zyx; xzy ) } K {5.2I 2 6xyz}

S* Ia3d 24(d)4. Ia3d(f) {.3.V*2z}

71 V* Ia3d 24(c)2. 22

2 12 1 ..FYlxxx 4(a). 3. xxx;

22. . FYlxxxSyz (xyz; lOl+(xyz;yzx;zxy ) }<

22,.P Y*lxxx 12 3 8(a). 3. I+[xxx;

22..P Y*lxxx3yz 12 3(c) I+{xyz;\0\+(xyz;yzx;zxy)}<

2 1 3.SVlz I2 13 12(b)2..

. . I 2 Y**lxxx 1+[xxx; J J J+ (xxx); ); IH+(xxx)

.3.J 2 S*V*lx Ia3 24(d)2..

Table 19: Hexagonal System

(and rhombohedral system in hexagonal description)

P P6/mmm l(a)6/mrm 000

P2z P6/mmm (e ) Oy 0^ ±z P3xx P6m2 (j ) xxO; x, 2x> 0; 2x> £, 0 P3x P6 2m (f) xOO; 0x0; xxO P6x P6/mmm (j ) ± (xOO; 0x0; xxO) P6xx P6/mmm (1) ± (xxO; x> 2x, 0; 2x, x, 0) ..2P 3x P6 Jmom (g ' ) xOO; 0x0; xxO; 00\+ (xOO; 0x0; xxO) ,2.PCE3xx P6 /mme (h ' )xxO; x, 2x> 0; 2x, x, 0; Q0\+ (xxO; x, 2x> 0; Zx, x3 0) 3 2 ..p£*Q2x P6 922(g) ±(xOO);00\±(xxO);00%±(OxO) 3 . . P +Q2xx P6"22 (i ' ) ± (xxO); 00±± (2x^xy 0); 00\±(x, 2x, 0) *. Lr lu P3xy P6(j) xyO;y,x-y,0;y-x,x,0 P3xx2y PT$m2 (I) xyO;y, x-y, 0; y-x^ x> 0; yxO; y-x> y> 0; x> x-y, 0

P3xx2z ^ *^J 9 * J ~™9 *^J *^J ~" P3x2z Po2m(i) x 0 +Z'0 x +z*x x +z P3x2y P$2m (j) (xyO; y> x-y\ 0; y-x, 5, 0) T P6xy P6/m (j) ± (xyO; y, x-y> 0; y-x, x, 0) P6xz P3 lm(k) ± (xOz; Oxz; xxz ) P6xxz P'Sml (i ) ± (xxz; x> 2x> z; 2x, x>z) P6x2z P6/mmm (n) ± (x^ Oj ±z ; 0, #, ±z; x9 x* ±z ) P6xx2z P 6/17tf787l(O ) +(x X ~^~Z*X 2x ~^~2 * 2x X ~^~z)

72 P6x2y P6/77mn(p) ± (xyO; y3 x-y, 0; y-x, x, 0) T * .^rOP oxyQvv P6c2(k') xyO'y x—y O'y—x x 0'00%+(yxO'y—x y 0*x x—y 0) ,2.P^E3xy P62e(h') xyO; y, x-y> 0; y-x> x, 0; 00%+ (yxO; xr-y, y, 0; £, y-x9 0) 2 . .P E3xy P6~/m(h r ) xyO; y, x-y, 0; y-x^ X, 0; 00%+ (xyO; y, y-x, 0; x-y, x, 0) 3 . . P^+Q2xy P66 (c') ± (xyO); 00l± (y-x,x, 0); 00\±(y,x-y, 0) 72,P 6xy P67mec(Z) ± (xyO; y, x-y, 0; y-xy x, 0) ;00%± (yxO; x-y, y, 0; x, y-x> 0) ,.2PC 3x2y P6 /mcm(j') {xyO; y^ x-yy 0; y-x, x> 0; 00%+ (xyO; y-x3 y3 0; x> x-y> 0)}i OT3 *5 vO •? • • £-£ Ol\.£.£i P6~/mcm(k r ) x> 0, ±z; Q> x, ±z; x> X, ±z; 00%+ (x> 0> ±z; 0, x> ±z; x, x> ±z ) {m. ,P6xz} .2.P E3xx2y xyO; y3 x-y, 0; y-x3 x, 0; yxO; y-x, c 3 y, 0; x, xr-y> 0; 00%+ (xyO; y, y-x, 0; xr-yy xy 0; yxO; x-y, y, 0; x> y-x, 0) .2.P E3xx2z P6 ~/nvnc (k ' ) x, Xj ±z; x, 2Xj ±z; 2xy xy ±z; 00%+ (x> Xj ±z; xs 2x, ±z; {m. . P6xxz} 2x x ~*~z )

P3xy2z PS(l) x, y, ±z; y, x-y, ±z; y-x> x, ±z P3xx2yz P312(l) xyz; y, x-y, z; y-x, x3 z; yxz; y-x, y, z; x, x-y, z P3x2yz P321 (g) xyz; y> xry, 2; y-x9 x, z; yxz; x-y, y> z; x3 y~x, z P6xyz P2(g) ± ( xyz; y, xry, z; y-x, x,z) P3xx2y2z P3m2(o) x, y> ±z; y, x-y> ±z; y-x> x, ±z; y> x, ±z; y-x> y, ±z;

P3x2y2z P62m(l) (x* y, ±z; y, x-y, ±z; y-x, x,±z)i P6x2yz P622(n) xyz; y> x-y> z; y-x^ x9 z; xyz; y^ y-x> z; x-y^ x, z; yxz; xry, y> z; x> y-x, z; yxz; y-x, y, z; x> x-y> z P6xy2z P6/m(l) ± (x, y> ±z; y> x-y^ ±z; y-x, x3 ±z ) P6xz2y P32m(l) + (xyz ' u x~y z • zy— x x z ) T P6xxz2y P'Sml(j) ± (xyz; y> x-y> z; y-x, x> z; yxz; x-y> y> z; x, y-x, z ) P6x2y2z P6/mmm(r) ± (x3 y> ±z; y, x-y, ±z; y-x, x3 ±z)i ..2P 3xy2z P6c2(l') x, y, ±z; y, x-y, ±z; y-x, x, ±z; 00%+ (y, x, ±z; y-x, y, ±z; {m. .P 3xx2yz} x>x-y,±z) .2.P E3xy2z P62c(i') ^ j, +z"u x~u +z*u—x x +z'00%+(y x +z'x-y y +z m {m, .P 3x2yz} X L/""^C +2 ) 30 . .Pp:*:Q2x2yz P6 922(k) XU Z * XLi Z *X~**Li ZV ? * 1J m**fr U ^ * C)0 TF+ (1J**mff* 7 P * T*"~7V T* ^ " 2 C Ct X3 y-x3 z; x3 x-y> z); 001+ (y3 x-y> z; y3 y-x3 z; yxz3 yxz) . , 2P 3x2yz P6 722(i) xyz; y, x-y3 z; y-x9 x3 z; yxz; x-y> y> z; x3 y-x3 z; {?2.P E3xx2yz} 00%+ (xyz; y> y-x> z; x-y3 x3 z; yxz; y-x3 y3 z; x3 x-y, z) 2 ..P E3xy2z P6Jm(i') x* y> ±z; y3 x-y3 ±z; y-x3 x> ±z; 00%+ (x3 y> ±z; y3 y-x> ±z; {m..Pc6xyz} o x-y3 x3 ±z ) . .cP E3xx2yz Piled') xyz; y> x-y> z; y-x, x3 z; yxz; y-x3 y3 z; xy x-y3 z; {7.2P 6xyz> 00%+ (yxz; x-y> y> z; x, y-x^z; xyz; y, y-x3 1; x-y, x, z ) .n.P 3x2yz P3cl(g') xyz; y, x-y> z; y-x9 xy z; yxz; x-y, y> z; x, y-x, z; {?2.Pc6xyz> 00%+ (yxz; y-x, y, z;x, x-y, z; xyz> y> y-x, z; x-y> xy z ) m. .P 6x2yz P6/mcc (m') xyz; y, x-y, z; y-x, x, z; xyz; y> y-x, z; x-y> x> z; yxz; {?2.P 6xy2z} x-y* y, z; x> y-x> z; yxz; y-x, y> z; x, x-y> z; 00%+ (xyz; y* x-y, z; y-x, x> z; xyz; y, y-x, z; x-y> x, z; yxz; x-y, y, z; x, y-x> z; yxz; y-x, y> z; x, x-y> z ) ..2P 3x2y2z Pt (x> y, ±z; y, x-y> ±z; y-x, x, ±z ) T; 001+ (x> y, ±z; y, y-x, ±z; {m..PG6xz2y} .2.P E3xx2y2z P6~/mnc(l r ) x> y, ±z; y, x-y^ ±z; y-x> x, ±z; y, x, ±z; y-x> y, ±z; x> x-y, ± {m..P 6xxz2y} 00%+ (y3 x, ±z; x-y> y, ±z; x, y-x, ±z; x, y> ±z; y, y-x, ±z;

P6/Tnmin 2(c)6m2

G2z P6/mmm(h)

73 E P6 /mmc 2(c)6m2 o ±(HJJ E2z P6 Jrrmo (f) c ±a,i,t±*)

R R3m 3(a)3m. 000;±(\\\)

R2z R~3m(o) R±(00z) R3xx R3m(b f ) R+ (xxO; x> 2x> 0; 2xy x>0) R3x R32(d) R+(xOO;OxO;xxO) R6x R3m(f) R±(xOO;OxO;xxO) .n. 'R 3x R3c(e') R+ { xOO; 0x0; xxO; 00\+ (xOO; 0x0; xxO) }

R3xy R3(b r ) R+ (xyO; y,x-y> 0; y-x, x, 0) R6xxz D-J- /MMM • M P*Y* j> • P^ y p ) R3xx2y R3m(c f ) R+ (xyO; y> x-y, 0; y-x> x, 0; yxO; y-x> y, 0; x, x-y, 0) .n. 'Rc 3xy R3c(b f ) R+{xyO; y> x-y> 0; y-x, x> 0; 00\+ (yxO; y-x, ys 0;

R3x2yz R32(f) R+ (xyz; y, x-y, z; y-x, x, z; yxz; x-y, y, z;

R6xyz rise/; R± (xyz; y> x-y, z; y-x,, x,z) R6xxz2y R± (xyz; y, x-y3 z; y-x, x, z; yxz; x-y, y, z;

.n. f R 3x2yz R+ {xyz; y, x-y> z; y-x> x, z; yxz; x-y> y> z; x, y-x, z; {.5. f R 6xyz} 00\+ (yxz; y-x> y> z; x> x-y, z; xyz; y, y-x, z;

N PB/mrnm 3(f)mmm \00;0\0; \\0

N2z P6/mm(i)

3(c)222

Q2z P6222(f)

R3m 9(e). 2/m. R+(\00;0\0;

.2.GElz P3ml 2(d)3.m

32 ..Pc T Qlxx P32 12 3 (a*). .2

3 2 ..Pc+ Qlxx2yz P32 12(o') 00 1+ (y-x, x> z; y-x, y>z)

32- .Pc+Qlx P32 21 3(a'). 2. xOO * Ox ^ * xx ~

3 r .PcT Qlx2yz P3221(c') xyz; x-y, y;_z; 001+ (y> x-y> z; yxz ) ; 001+(y~x*Xjz;x,y-x,z)

74 3 I2PCc Qclx 6 (a).2. xOO;xxl;Ox\;xO\;xx\;Oxl 3r 2Pc >x~y* z;yxz); 00\+ (xyz;y-x,y,z); 00\+ (y~x,x> z; y-x, z);00?+(yj y-x, z; yxz )

P F f) 1 w P(^ 99 ••' «X'*OU'^ £j»X/_j ^^ 6 ,J «^> LJtJv } 3 J «^«^ 2 J

3,- • Prc hQlxy ^^2PQ 3fa f ;^ ^^.^ y^y 2 ^ £ J

3i2r .P EC+Qc lxy P6 1 6(a')l /^ i;^J; ~ 2 5

Table 2o: Rhombohedral complexes in rhombohedral description

P P3m l(a)3m 000

P2xxx P3m(c) ± (xxx) P3y P3m(b r ) OOZK P3xx P32 (d) P6xx P3m(f) ± (xxO) K .nI3xx P3c(e')

P3yz P32(b') P6xxz P3m(h) ±(xxz)< P3y2z P3m(c') Oyzi\ .nI3yz P3c(b') {Oyz; \\\+(Ozy) }K

P3xx2yz P32(f) P6xyz P31(f) ±(xyz)< P6xxz2y PZm(i) ±(xyz)i\ ,n!3xx2yz P3e(f) {.2I6xyz}

J P3m 3(e).2/m \0\<

75

501-563 O - 73 - 6 Table 21: Tetragonal System

P P^/mmm l(a)4/mmm 000 P2z P4/mmm(g) ±(00z) PHxx P4/mmm(j) ± (Xj ±x, 0) P4x P4/mmm(l) ±(xOO)i .b.C2xx P4/mbm(g f ) ±(xxO);^0±(xxO) ..2P I2x P42/mme(j) ±(xOO;Ox\) .2.PC2xx P4 /mcm(i) ±(xxO;xx\) .b.C°2xx I47mcm(h r ) I±(xxO;xx\) c PUxxz P42m(n) xxz;5cxz;xxzi: P4xz P4m2(j) ±x> 0, z; 0, ±Xj z P4xy P4/m(j) ±(xyO;yxO) P4x2y P4/mmm(p) ±(xyO;yxO)n P4xx2z P4/mmm(r) ± (Xj Xj ±z; xs Xj ±z ) PUx2z P4/mmm(s) ±(XjO;±z)i: 5 . .P I2xy P4jm(j) ±(xyO;yx\) .2.P°4xy P4/moe (m) ± (xyO; yxO; xy I; yx\ ) . . 2c5xxz P4/nbm(m') xxz; xxz; xxz T; \\0+ (xxz; xxz; xxz T ) . .mC^xy P4/mbm(i) ±(xyO;yxO); \\0±(xyO;yxO) {.b.C2xx2y} .b.C2xx2z P4/mbm(k r ) ± (XjXj ±z); \ \0± (x> x> ±z) . . mCHxz P4/nmm(i) ±x> 0> z; 0, ±X; z; \\0+ (0, ±x, z; ±x* 03 z) {..2CIlz4x} m. .P 4xz P4 /mmc(o') ±x, 0; z; 09 ±Xj z; 00\+ (±x^ 03 z; 0, ±x^ z ) {?.2P I2x2z} ..2P I2x§y P4 /mme(q) ± (xyO; xyO; yx\; yx\ ) .2.P°2xx2y P4 9/mam(n) ± (xyO; xy\)i m. .PC4xxz P4p/mcm(o r ) xxz; xxz; xxz i; 00\+ (xxz; xxz; xxz T ) {?2.P 2xx2z> .b2C 2xy° P4JmbQ(h) ±(xyO);llO±(3yO);OOl±(yxO);\l\±(yxO) . ,mC CHxy 14 /mom (k) I± (xyO; yxO; xy \; yx\) {?2.C 2xx2y} .b.C Hxxz 14/mcmd 1 ) I+{xxz;xxz;xxzn; \lO+(xxzi;xxz;xxz) } {?b.C 2xx2z} c PM-xyz P4(h) xyz; xyz; yxz; yxz PHxxz2y P32m(o) (xyz; xyz; yxz; yxz ) T P4xz2y P4m2(>Z) x, ±y, z; x, ±y> z; y, ±x> z; y, ±x, z P4x2yz P422(p) xyz;xyz;xyz; xyz;yxz; yxz; yxz; yxz P4xy2z P4/m(l) - (%* y* -s j y 3 *c> -s J P4x2y2z P4/mnm(u) ± (x*yj±z;yjXj±z)t .2.Pc4xyz P42c(n) xyz; xyz; yxz; yxz; 00\+ (xyz; xyz; yxz; yxz ..Pc I2x2yz} •mCUxyz P42 2m(f) xyz;xyz;yxz;yxz; H0+(yxz;yxz;xyz;xyz) {.2 1 .CIlz2xx2y} .2Pc4xyz xyz; xyz; yxz; yxz; 00\+ (yxz; yxz; xyz; xyz) {5 1 ..? 2xx2yz} xyz; xyz; yxz; yxz; \ \0+ (yxz; yxz;xyz; xyz ) {.b.C2xx2yz} .2 .C2xx2yz P42^2 (g) xyz; xyz; yxz; yxz; I \0+ (xyz; xyz; yxz;yxz) {..2CIlzHxy}

76 ..2PcI2x2yz P4 922(p) & xyz; xyz; xyz; xyz; 00?+ (yxz; yxz;yxz; yxz ) {.2.Pc 2xx2yz} m. .P£4xyz P4jm(k') xyz; xyz; yxz; yxz; 00\+ (xyz; xyz; yxz; yxz ) {U 1 ..Pc2xy2z} Ci lC4xyz P4/n(g) xyz; xyz; yxz; yxz; l\0+ (xyz; xyz; yxz; yxz ) •f *| O T 1 r* \\ Vt T \- 1 -LV^J. JLZiT"AV J P4/mee(n') xyz; xyz; xyz; xyz; yxz; yxz; yxz; yxz; {?2.P 4xy2z} 00\+(xyz;xyz;xyz;xyz;yxz;yxz;yxz;yxz) . ,mC4x2yz P4/ribm(n) xyz;xyz;xyz;xyz;yxz;yxz;yxz;yxz; {. . 2C4xxz2y} \\0+ (yxz; yxz; yxz; yxz; xyz; xyz; xyz; xyz) . .mC4xy2z P4/mbm(l) ± (x>y* ±z;yjX* ±z) ; ??0±(y>Xj ±z;x^y^ ±z) {.b.C2xx2y2z} . .mC4xz2y P4/nmm(k) x +y Z'x +y z"u +x z'ij +x z'\\0+( +y x z* {..2CIlz4x2y} +u x Z'+x M z'+x y z) . . c2C Hxyz P4/nco(g) xyz;xyz;yxz;yxz;^20+(xyz;xyz;yxz;yxz); {?2 1cC 2xx2yz} 00\+ (yxz; yxz; xyz; xyz ) ; \\ ?+ (yxz; yxz; xyz; xyz) {1.2(cSlz) 4xy} m. .Pc4xz2y C P4 /mmo(r 1 ) x, ±y^ z;x> ±y^ z;y, ±x> z; y> ±x> z; 00\+ (x> ±y> z; {. .2P I2x2y2z} X +U Z'lj +X Z ' 11 +X Z ) m. .P Hxxz2y P4 /mem(p r ) (xyz;xyz;yxz;yxz)i;00\+(xyz;xyz;yxz;yxz)i {?2.P 2xx2y2z> £j .22C 4 xyz P4 /nbc(k) xyz;xyz;yxz;yxz; \\0+(yxz;yxz;xyz;xyz) ; {?b2C 2x2yz) 001+ (xyz; xyz; yxz; yxz); H H (yxz; yxz; xyz; xyz) {.2cC C2xx2yz} m.2C 4 xyz P4jmbe(i' ) xyz;xyz;yxz;yxz; ??0+(yxz;yxz;xyz;xyz); {?b2C 2xy2z> Ci 00\+ (xyz; xyz; yxz; yxz );???+ (yxz; yxz; xyz; xyz) {mb.CG2xx2yz} . .m2C 4 xyz P4jnom(j) xyz; xyz; yxz; yxz; \\0+ (yxz; yxz; xyz; xyz); LJ {?2 mC 2xx2yz} ^Hf^i;^i;^ {4..F2xxz2y} {.2 12(CIlz) c 2xx2y} . . 2C 4xyz I4o2(i) I+{xyz;xyz;yxz;yxz; \?0+ (xyz; xyz; yxz; yxz) } {?b.C 2xx2yz} .b.C 4x2yz 14 /mom (m ' ) I+{xyz;xyz;xyz;xyz;yxz;yxz;yxz;yxz; {?b.C 4xxz2y} 00\+ (xyz; xyz; xyz; xyz; yxz; yxz; yxz; yxz) } {.b.CC4xy2z> {.b.C C 2xx2y2z}

I lU/mmm 2(a)4/mrrm 000;???

I2z I4/mmm(e) I±(00z) mxx I4/mmm(h) I± (x, ±x> 0) IM-X I4/mmm(i) I±(xOO)i ,n.I2xx P42/rmm(f) ±(xxO);\^±(xxO) .2.F 2xx 14 /ciccKi^ ) 1+ \ — ( xxO ) ' ~o ~2 0— ( xxO ) * ~2 0 u— ( xxO ) * 0? ii— ( xxO ) / c 1 I4xz I4m2(i) I+(±x>0,z;0>±x,z) I4XXZ 14 2m (i ) 1+ (xxz; xxz; xxz T ) I4xy I4/m(h) I±(xyO;yxO) I4x2y I4/mmm(l) I±(xyO;yxO)i I4xx2z I4/mmm (m) I± (Xj Xj ±z; x, Xj±z) I4x2z I4/mmm(n) I±(x,03 ±z)i . ,2I4xy P4/rme(h) ± (xyO; yxO) ;???± (yxO; xyO) xxz;xxz;xxzt; .n.I2xx2y P42/rrmm(i)

77 .n.I2xx2z P4r>/mnm(j) ±(x^x3 ±'t ^.cIUxz P4,>/nmc(g) ±x>0,z;( *+..F2xxz P4jncm(i') ±(xxz); {.2 2(CIlz) 2xx) ^>xy ° I4 2cd(b r ) 7+ {±(xyO); \\0±(xyO); \Ol±(yxO) ;0^±(yxO)}

I^xyz I4_(g) I+(xyz;xyz;yxz;yxz)

I42m(j) J+ (xyz;xyz;yxz;yxz) T I4x2yz 1422(k) 1+(xyz;xyz;xyz;xyz;yxz;yxz;yxz;yxz) J.T LL*T VXT'^V O^- "7/-i -*-7"4 J //7771 ''« Itx/ •"? ) JL—7~~i~ IT*( *b j y?v 3 ~^~—" ? y* nu 3 T* y ~^~ "z* /) I4x2y2z I4/mmm(o) I±(xjt y3 ±z;y^x^±z)i . .cl^xyz P42 c(e) xyz;xyz;yxz;yxz; \\\+(yxz;yJ£z;xyz;xyz) . .2I4xyz P4nz(i) xyz;xyz;yxz;yxz; ±^+(yxz;yxz;xyz;xyz) {.n.I2xx2yz} .2 .I2xx2yz P4 2 2(g) xyz;xyz;yxz;yxz;lll+(xyz;xyz;yxz;yxz) n.. I4xyz P4 /n(g) xyz;xyz;yxz;yxz; \\^(xyz;xyz;yxz;yxz) .. cI4x2yz P4/nno (k) xyz; xyz; xyz;xyz; yxz; yxz; yxz; yxz; 2 i 2 + ("2/0:2; ^5s; 2/^3; ^xs; 5^s; 5^/s; xz/s; o^s J

. .2I4xxz2y P4 /nnm(n) (xyz;xyz;yxz;yxz)i; \\l+(yxz;yxz;xyz;xyz)i .n. I2xx2y2z P4 /rrmm(k) ±(x^y^±z)i; l^±(x^y^ ±z)n .. cI4xz2y P42/nmc(h) x> ±y, z;x> ±y,z;y,> ±x^z;y^ ±x>z; \\%+(y> ±x>z;

.22F 4xyz I4^/aed(g) xyz;xyz;yxz;yxz; \\0-\-(yxz;yxz;xyz;xyz); (?b2.Fc 2xx2yzl ^0^(xyz;xyz;yxz;yxz);OH+(yxz;yxz;xyz;Syz)

D F4 1 /ddm 8(a)42m F+(000;lHi)

VD 14 /amd 4(a)4m2 1+(000; OH)

vD2z I4~/amd(e) I^(0^0^±z;0^\^l±z) VD4xx 14 /amd(g) I+{±(Xj ±x> 0);0%%±(x, ±x, 0)} ..dVD2x 14 md(b') I+{±(xOO);OH±(OocO)} .2. VD2xx I4 2 22(d)

4j./D2xy I4 2 (b') .2. D4xz I4^/amd(h) I+{±XjOjZ;Oj±XjZ;OH+(±XjOjZ;( . .dVD2x2y I4^r».d(c r ) I+{±(Xj ±y^0);0\^±(±y3 x^0)}

.2. VDUxyz I42d(e) .2/D2xx2yz 14 ^2 (g) a. . D4xyz I4~/2(f) I-\-{xyz;xyz;yxz;yxz;OHjr(xyz;xyz;yxz;yxz)}

2 • T\\\l/T"A.^^iy V79\7 _t7"4 tr — //HYflrf LAjllU, (1It'/ ) .17~4- i -f\ *iLT™ « +7y—^ j ^_jJ' * "O^* +7V^73 '••'3/73^ * 77 -f—«^3 O* ^3/735 * f> +'Y*—*i'3 '^35 • ^*• J. — — — i -, oj x2V Z * U —X Z * *U "• 1* Z ) \

. .2/m

/amd 8(c).2/m.

..2 VT2x I4 2/amd(f) 1+ (±xf J, i; ic, I, i; i, ±or, I; I, ±x, I)

78 .2CIlz 2(c r )4rm

,2 .CIlz2xx xxz; xxz; 110+ (xxz T ) ..2CIlz4xx P4/nmm(j') XXZ;XXZ;XXZT;

4-..P I Ix P4 0 22 4(a'). 2. xCO'Ox^-xOl •Or 1 3 cc c 3 ^ 3 ..P I Ix2yz P4 -22 (d r ) xyz; xyz; 001+ (yxz; yxz); 00%+ (xyz; xyz) » •ftt "D F)1'VV'?\7''7r \ • n • • -L U J-f^i^sLy Jtj J 001+ (yxz;yxz) O CC

^3" P VDlxx PU 322 4(0' / . . ci XX U _j XX cc i^ ; XX ~2 y XX if

4 0 . .1 VDlxx .2 xxO; 3 c pVi2 4 (a). 2 51*+ (xxO) ; xx\;

..I Dlxx2yz xyz; yxz; J H+ xyz); 00%+ (xyz; yxz) ; lll+(yxz;xyz)

T+ /^.ll . «31 . 3^7. 1-7 Ix J.T I *L i+ ej «^ 14 8 j ***^ 8j U*^ 8

TJ./ r l 1.^31.3 = 7. 1_7 .22 TC Ix J.T cc 8(f).2. ( Xu 8 ^ X 14 8 j U»C8j t^^S

I^/acd 1i+j 1-^i+ ^^ JL.3i+^ i+j uTX_,3, 1.3i+ 3 14^ 1,«^T*^'J 1.u_> u_, ^""X^ ij/

0 ..P VDI Ixy 4(a')2 3 cc c y

Table 22: Orthorhombic System

P Pmmm I (d)mmm 000

P2x Pmmm(j) ±(xOO)

P2y2z Pmmm(u) ±(0,y,±z) Peom(q) ±(xyO);±(xyl) m..P 2xz Prrma(i) ±(xOz);\00±(xOz) {f2.PaBlz2x} .m. Pab2yz Crrma(m) C±(0yz;\yz) {2..PabPlz2y}

79 P2x2yz P222(u) xyz; xyz; xyz; xyz P2x2y2z Pmmm(a) ±(xy y,±z;x>y>±z) ..mP 2x2yz ) xyz;xyz;xyz;xyz;00^(xyz;syz;xyz;xyz) {5..PG2xy2z} m..Pa2xz2y Prrma(l) ±(x, ±y>z); \00± (x, ±y,z) {.2.P Blz2x2y} .22P 2xfz Pooa(f) ±(xyz);\00±(xyz);00\±(xyz);lOl±(xyz) {c.2P Fly2xz} {12.(f°Blz) 2xy) 2.mP 2xyz ° Pbom(e) ±(xyz);0±0±(xyz);00\±(xyz);0l\±(xyz) {.^(P Clx) 2yz) {2.1P, A.C Flxy2z) m..P K 2x^? ° Cmrna(o') C-^{xyz;xyz;xyz;xyz;^00^(xyz;xyz;xyz;xyz) } (ffe.P K 2yz2x] {m..PaT2xz2y} {2..PafFlz2x2y} 22.P 2xyz Iboa(f) I^{±(xyz);^00±(xyz);0l0±(xyz);00l±(xyz)} {T22(P Fix) 2yz}

C Cmmm 2(a)rmm 000; \\0

C2x Crrmn(g) C± (xOO) C2z Ommm(k) C±(00z)

C2y2z Cnwvn(n) C±(03 y,±z) C2x2y Cnvnrnfp) C±(x,±y3 0) .2.B2yz Pmna(h) ±(0yz);\0\±(0yz) b..C2xy Pbam(g) ±(xyO);\\0±(xyO) .n.C 2yz Cmom(f) C+{±(Oyz);00\±(Oyz)} {5..C Fly2z} n..C F2xy CccmCl) C±(xyO;xy\) b. .C C 2xy Ibam(j) I±(xyO;ocyi) .2.BSyz Imma(h) I+{±(Oyz);OlO±(Oyz)} {72. B, A Iz2y} D a. C2x2yz C222(l) C+ (xyz; xyz; xyz; xyz) C2x2y2z Cmmm(r) C± (Xj y> ±z; x^y^±z) b. .C2x2yz Pban (m) xyz; xyz; xyz; xyz; 1 J0+ (i 2.2A 2xyz Pnna(e) ±(xyz) ;0\\±(xiiz) -$00+ {f/n. .B A Iz2xy)

,2.B2yz2x Pmna(i) ±(±x,y,z);\0\±(±x,y,z, b. .C2xy2z Pbam(i) ±(xsyj±z);^0±(x>y,±z, b2.C G 2xyz Pbon(d) ±(xyz); \\0±(xyz);00:2 ± {b.2 1C GFly2xz} .maBj?2xyz Pnma(d) -L-\- ({ Juyf+jj <5y /) •* 12^2 O— —•+• *f ***yT*77 &5 /) j• ^^0— 2 ^/0+ — {l^RA FI Ixz2y} .n.C 2yz2x a a Cmem(h) C-\-{±(±x^y3 z) ; \\\±(±Xji n. ,C°2x2yz Ccem (m ' ) C+{xyz;xyz;xyz;xyz; H" {n..C F2xy2z} b..C 2x2yz Ibam(k') I+{xyz;xyz;xyz;xyz; Ji{ {6..C 2xy2z)

80 .2.B 2yz2x Irrma(j) I+{±(±x,y9 z);\0\±(±x3 y9 z)} {2..A 2xz2y} {.2.B^A Iz2x2y}

Immm 2(a)mmm

I2x Immm(e) I±(xOO)

I2y2z Irrmn(l) I±(0,y,±z) n.,I2xy Pnnm(g) ±(xyO);\\l±(xyO)

I2x2yz 1222(k) J+ (xyz; xyz; xyz; xyz) I2x2y2z Irrswn(o) n..I2x2yz Pnnn (m) xyz; xyz; xyz; xyz; H s+ (xyz; xyz;xyz; xyz ) n..I2xy2z Pnnm(h)

F Fmmm 4(a)mmm 000;0l\<

F2x Fmmm(g) F± (xOO)

F2y2z . 2.F2yz Cmca(f) C+{±(Oyz);\0\±(Oyz)}

F2x2yz F222(k) F+(xyz; xyz;xyz; xyz) F2x2y2z Fmmm(p) c.2F2xyz Pccn(e) ±(xyz);0\\±(xyz);\0\±(xyz);\±0±(xyz) 2xy} bc.F2xyz Pbca(c) ±(xyz);0\l±(xyz);\0\±(xyz);\lO±(xyz) .2.F2yz2x Cmoa(g) c. .F2x2yz Ccca(i) C+{xyz;xyz;xyz;xyz;lO\+(xyz;xyz;xyz;xyz)}

Fddd 8 (a) 222 F+(OQO;IW

D2x Fddd(e)

d..D2xy Fdd2(b')

d. .D2x2yz Fddd(h)

Fddd 16(o)J

.2.P Biz Pmma 2(e')mm2

.2.P Blz2y Pmma(k') 3. ,2.PcBlx2yz P2222 (e) xyz; xyz; 00%+ (xyz; xyz) {2..P Aly2xz}

81 2 r .CIlz Pinmn

.CIlz2y Pmmn(e)

2 ..CIlz2xy xyz;xyz; 2^..CIlz2x2y *» ty* z;£j ±j/, z; H0+ fa, ±y,5;£, ±j/, 5;

Cmcm 4(c')m2m C+(OyO;Oy\)

2 ..C Fly2x C'mam (g')

.2 .C Flx2yz (5..C Fly2xz)

2. • PabFlz Cmma 4(g')mm2 C+(OOz;0\z)

. V Iz Imma 4(e')mm2 b a ,2C Bb lx2yz I+{xyz;xyz; l? {2..A C Iy2xz} {.2.B^AC lz2xy} JD a

Pbcm 4(d')..m

1.2,B, A FI Ixz Pnma *Ov-/rfi7 12i2'^ lj-» 1 b a a 4(0') .m.

,2,.FA B^C I LI Ixyz P2,2,2, 11 abcabc-7 111 4(a)l xyz; l-Xjy* i+s; 1+a;, J-j/, s;

Table 23: Monoclinic System p P2/m l(a)2/m 000

P2y P2/m(i) ±(OyO)

P2xz P2/m(m) ±(xOz)

P2xz2y P2/m(oJ ±(x,±y,z) TnP^2xyz P2 1/m(f) ±(xyz);0%0±(xyz) {2 1PbACIlxz2y} 2P 23QTZ P2/e(g) ±(xyz);00l±(xyz) {cP Aly2xz}

82 c C2/TT1 2(a)2/m 000; \\0

C2y C2/m(g) C±(OyO)

C2xz C2/m(i) C±(xOz)

C2xz2y C2/m(j) cA2xyz P2 /c(e ±(xyz);0\\±(xyz) 2 CfiF2xyz C27c(f) C+{±(xyz);0\\±(xyz)} {1C Fly2xz} cP Aly P2/c 2(e r )2

C2/c C+(OyO;Oyl)

Ib1 ^ 2(e')m

Table 24: Anorthic System

PI 000

P2xyz Pl(i) ±(xyz)

83

Occurrence of Lattice Complexes Table 25

Lattice complex Multi Site Occurrence : in all its plicity symmetry space group, Wyckoff letter, representations (oriented) shift from standard representation

Cubic, invariant p 1 m3m 43m P43m(a) 3 (b)\\\ 432 P432(a)> (b) Hi m3. Pm3 (a) 3 (b ) Hi 23. P23 (a),(b)\\\ ( a) P2 8 43m Fm5m(c)V*l 432 FmSc(a) III. m3. FmSa(b) 23. moM.wu

Fm3 (c) J J i . 3m Tm3m(c) uuu .32 Pm3yi ( 6- ) u u u

.3. ^rlnli J7773 f<3>) i Ji Ja3 Tajj (b)^H

I 2 m3m Im3m(a) 43m Pn3m(a) 1 4 3m (a) 432 Pn3n(a) 1432 (a) m3. Pm3n(a) Im3 (a) 23. P43n(a) P4 232(a) Pn3 (a) 123 (a)

( a )200/020/002. From standard representation to cell.

85 (a) I 16 23. Fd3e(a) .3. Ia3d(a)

0 3 4/mm.m PmZM.Wm 42. m P43m(c) j (d)\\\ 42.2 P432(c) j (d)\\\ mmm. . Pm3 (c) 3 (d) \\\ 222.. P23 CeVfdJHi ( a) J 24 4m. 2 Rn3c(c)Hl 4/m.. Fm3a(d) 4.. F?3c(c) t (d)Hl m. mm Fm3m(d) 2.22 F432(d) 2/m. . Fm3 (d)

0 4 m3m ^m(a),(b)\\\ 43m F43m(a), CbJ Jik ^441, (d)Hl 432 F432(a) 3 (b)m m3. Fm3 (a), (b)l\\ 23. F23 (a) f (b)in,(c)Ml,(d)lH .3m Pn3m(b)lH,(c)Hl . 32 P4 g 32(b)Hlf (c)iH .3. Pn3 (b)lU,(c)ttl Pa3 (a) , (b)^\\ ( a ) F2 32 .32 Fd3a(b)lll .3. Fd3o(o)\U

E3 4 .32 P4332(d) t (b)M (b ) "Y 4 .32 P4^(a),(b)^

0 6 4m. 2 PmZn(c),(d)^ 4.. P43n(c)\\\,(d) 2.22 P423SM,(f)M

j* 6 4/mm.m Im3m(b) 42. m Pn3m(d)

( a )200/020/002. (b )100/010/001. .

86 I43m(b) 42.2 Pn3n(b) 1432 (b) mmm. . Pm3n(b) Im3 (b) 222.. P43n(b) P4 232(d) Pn3 (d) 123 (b) ra) j*2 48 4.. Fd3c (d)

0 8 43m Fd3m(a),(b)m 23. F4 1 Z2(a),(b)\H Fd3 (a) , (b)l\\

+Y* 8 .32 I4 2 32(a) ( b ) ~Y* 8 .32 I4 2 32(b)

|w*| 12 4m. 2 Im3m(d) 4.. Pn3n(d) I43m(d) 2.22 Pn3m(f) 1432 (d)

12 4.. I43d(a)

12 2.22 I4 2 32(e) (b ) "v 12 2.22 I4 2 32(d)

0 16 .3m Fdlm(c),(d)\\\ .32 F4 1 32(c),(d)Hl .3. Fd3 (c) 3 (d)\\\

IY**] 16 .32 Ia3d(b)

( a )200/020/002. (b )100/010/001.

87 s* 24 4.. Ia3d(d)

|v*[ 24 2.22 Ia3d(c)

Cubic, univariant

P4xxx 4 . 3m P33m(e) .3. P23(e) (*) (P,xxx) 2 32 .3. mc(e)

22. .FYlxxx 4 .3. P22 S(a)

P6z 6 4m. m Pm3m(e)j (f)\\\ 4.. P432(e) (f)~\\ mm2. . Pm3 (e) j (h) m 2. mm P42m(f) s (g)\ll

2.. CP91 £j*J ^( J-P) ' J '(-1 ^ ') 1J222- ^ ( a ) (P6z) 2 48 4.. Fm3a(f)m mm2. . Fm3c(e) 2. mm Rn3m(g)Hl 2.. n*>(f),(g)M

jP777oTTVvi "2 (/ gxv )\ uu1 1 u1

.3.J2x 6 mm2. . *n*

P8xxx 8 . 3m Pm3m(g) .3. P432(g) Pm3 (i) ( a ) (PSxxx) 64 .3. Fm3c(g)

. . 2l4xxx 8 .3m Pn3m(e) .3. P4232(g)

( )200/020/002.

88 Pn3 (e) ( a ) (..2IHxxx) 2 64 .3. Fd3c(e) s 8 . 3m aam(c) .3. P43n(e) 123 (o)

4 3 .. +Y2xxx 8 .3. 8 .3. P4 2 32(o)

be. .F2xxx 8 .3. Pa3 (c)

2 12 r .P2Y*lxxx 8 .3. I2 2 3(a)

.3.J*x 12 mm2. . Pm3m(h) 2.. P43m(h) P432(h)

P12xx 12 m. m2 ..2 S'w!!! ( a ) (P12xx) 2 96 ..2 FKtoMlll

,3.W2z 12 mm2. . Pm3n(g)> (h)\\\ 2.. P43n(g)H?j (h) P4p32(i)j (j)\\\

I6z 12 4m. m Im3m(e) 4.. Pn3n(e) 1432 (e) rm2.. Pm3n(f) Im3 (d)

2.17871 Pn3m(g) I43m(e) 2.. P43n(f) P4 232(h) Pn3 (f)

( a )200/020/002. (b )100/010/001.

89 123 (d) 96 2.. Fd3c(f)

(333 12 U U 1+

12 ..2 P4 732(d) ,. . Y3xx t5 . . Y3xx 12 P4 2 32(d)

12 mm2. Im3 (e) Pn3 (g) 123 (e)

12 2..

16 . 3m Im3m(f) .3. Pn3n(f) Pm3n(i) I432(f) Im3 (f)

16 , 3m F43m(e) ,3. F23 (e)

4..I Y**lxxx 16 3. I43d(c)

16 ,3. I4 2 32(e)

16 .3. Ia3 (o)

24 ..2 Pm3n(j)

24 ..2

24 4m. m Fm3m(e) 4.. F432(e) mrn2. . Fm3 (e) 2.mm

( a )200/020/002, ( b )100/010/001.

90 2.. F23 (f),(g)Ul

,3.J*4;< 24 mtn2. . Im3m(g) 2.. Pn3n (g) Pn3m(h) I43m(f) 1432 (g)

1 12 xx 24 m.m2 Im3m(h) ..2 Pn3n(h) 1432 (h)

,3.S2z 24 2. . I43d(d)

U. .P 2 3xx 24 ..2 I422(i)lll

.3,V2z 24 2.. I4 2 32(f)

U . . +Y*3xx 24 ..2 I4 2 32(h) (»),, . . Y*3xx 24 ..2 I4 2 32(g)

.3.J2 S*V*lx 24 2. . Ia3 (d)

FSxxx 32 . 3m Fm3m(f) .3. F432(f) ^ Fm3 (f)

. . 2DHxxx 32 .3m Fd3m(e) .3. F4 2 32(e) Fd3 (e)

5. .Y**2xxx 32 .3. Ia3d(e)

F12xx 48 m.m2 FmZm(h), (i)\\\ ..2 F4Z2(g),(h)\l\

( b )100/010/001

91

501-563 O- 73 - 7 D6z 48 2. mm. Fd3m(f) 2.. F4 2 32(f) Fd3 (f)

4. .P 6xx 48 ..2 ImZm(i)lll

.3.S*2z 48 2.. Ia3d(f)

5a. .Y**3xx 48 ..2 Ia3d(g)

22..T3xx 48 ..2 F4 2 32(g)

5. ,T6xx 96 ..2 Fd3m(h)

52. .F 3x3 96 ..2 Fd2c(g)lll

Cubic, bivariant

P6z2xx 22 ..m P43m(i)

P6z2x 12 m. . M W.fWH.

P6z4x 24 m. . Brt.fW.fWHl

P6z4xx 24 . .m Pm3m (m)

..2I6z2x 24 m. . Pm3n(k)

..2I6z2xx 24 . .m Pn3m(k)

I6z2xx 24 . .m I43m(g)

I6z2x 24 m. . Im3 (g)

92 I6z4x 48 rn. . Im3m(j)

I6z4xx 48 . .m Im3m(k) * F6z2xx 48 . .m F43m(h)

F6z2x 48 m. . Fm3 (h)

F6z4x 96 m. . Fm3m(j)

F6z4xx 96 . .m Fm3m(k)

..2P2 6z2x 96 m. . Fm3c(i)

. .2D6z2xx 96 . .m Fd3m(g)

Cubic, trivariant (site symmetry 1)

The 12-pointers: Pm3 (I) P6z2xy P23 (j) P6z2x2y

Pn3 (h) 22, . FYlxxxSyz P2 2 3(b) n. .I6z2xy 1

Pa3 (d) The 24-pointers: be. .FBxyz

I6z2xy 123 (f)

2 2 ..P 9 Y**lxxx3yz I2 2 3(o) j_ j_ ^-

The M-8-pointers: P6z4x2y Pm3m(n)

n. ,I6zHxy Pn3n(i)

..2I6z2x2y Pm3n(l)

(b )lOO/010/001.

93 , . 2I6z2xx2y Pn3m(l) . ,2D6z2xx2y FdSm(i)

I6z2xx2y I43m(h) d.2I 26z2xy Fd3e(h)

,3dS4xyz I43d(e)

I6z4xy I432(j)

22. .Y*3xx2yz U^H>

I6z2x2y Im3 (h)

22. ,P2 6xyz Ia3 (e)

F6z2xy F23 (h)

I6z*4x2y Im3m (I)

5a..Y**3xx2yz Ia3d(h)

The 96-pointers: F6z2xx2y F43m(-i)

. .nP 6z2xy F43o(h)

F6z4xy F432(j)

. .2D6z2xy F4 2 32(h)

F6z2x2y Fm3 (i) d. .D6z2xy Fd3 (g)

The 192-pointers: F6zM-x2y Fm3m(l) m. .P 6z4xy «.«;,,.

94 Occurrence of Lattice Complexes Table 26-27 Lattice complex Multi Site Occurrence : in all its plicity symmetry space group, Wyckoff letter, representations (oriented) shift from standard representation

Hexagonal and rhombohedral, invariant

0 1 6/mmm P6/m™ (a),(b)OOl 6m2 P6m2(a)> (b)00\> (c)\\Q, (d)\\\>

62m P62m(a),(b)00l 622 P622(a),(b)00l 6/m. . P6/m(a), (b)OOl 6.. P6 fa^fWOOh^i^fdJHl,

3m. Plml(a), (b)00\

3.77? P31m(a) , (b)00\ 32. P321(a),(b)00l 3.2 P312(a), (b)OO^ (c)HO, (d)\\\,

3.. P3 (a), (b)00\ P[z] 1 6mm P6mm(a) 6.. P6 (a) 3m. PZml(a) y (b)\\0> (o)\\0 3.m P31m(a) 3.. P3 (a),(b)\\0> (c)\\0 < a > Pc 2 6m2 P63/mmc(b)00^ 62m P6 3/mom(a)OOl 622 P6 /nice (a) 00 1 6/m. . P6/mcc(b) 6.. P6c2(b)00l,(d)m,(f)m P62c(b)00l

1m. P6 3/mmc(a) 3.m P6 /mom(b)

( )100/010/002. From standard representation to cell.

95 32. P62c(a) P6 722(a) u P3cl(a)00l 3.2 P6-22(b)OOl o P31c(a)00l 3.. P6 3/m(b) P3cl(b) P32e(b) 6.. P6cc(a) 3m. 3.m 3.. P62 (a)

P31e(a) 222 , (b)OOl

2.. P6 2 (a) P64 (a)

6m2 P6/nrnn(o), (d)00\ 6.. P62m(c), (d)00\ P6/m (c), (d)00\ 3.2 P622(c), (d)00\

G[z] 3m. P6mm(b) 3.. P6 (b) P31m(b) 6.. P6/mcc(d) P6~/mcm(c)OOli

> (d)-00\

( a )100/010/002.( b )100/010/003.

96 6.. P62c(c),(d)00\

3.2 P6 22(c), (d)00\ PZ2c(c),(d)00\ E[z] 2 3m. P6jnc(b) o 3.. P63 (b) P31c(b)

R 3 3m. R3m (a) > (b)00\ 32. R32 (a), (b)00\ 3.. R3 (a), (b)OOl R[z] 3 3m. R3m (a) 3.. R3 (a) (°) 'Rc 6 32. R3c (a) 001 3.. R3c (b) ( C )

3 mmm P6/mmm(f), (g)00\ 0 222 P622(f),(g)00\ 2/m. . P6/m(f),(g)00\ .2/m. P'Sml(e)^ (f)00\ ..2/m P31m(f),(g)00i 1 P3 (e),(f)00\ N[z] 3 2mm P6mm(c) 2.. P6 (c) ( a ) Nc 6 222 P6/mcc(f)OOl 2/m.. P6/moc(g) .2/m. P63/mmc(g) . .2/m P6 3/mom(f) 1 P6 3/m(g) P3el(e) P31c(g) ( a ) Nc [z] 6 2.. P6oc(c)

+Q 3 222 P6 22 (c), (d)OOl +Q[z] 3 2.. P62 (b) ( d) "Q 3 222 P64 22(c),(d)00\

( a )100/010/002. ( C )010/110/002. ) 100/0 10/001.

97 ( d ) -Qfz] 3 2.. P64 (b)

M 9 .2/m. R3m (d)OOl, (e) I R3 (d)00\, (e) ( C ) »Mc 18 1 R3c (d)

Hexagonal and rhombohedral, univariant

P2z 2 6mm P6/mmm(e) 6.. P622(e) P6/m(e) 3m. P6m2(g),(h)\lO,(i)HO P3ml(c) 3.m P62m(e) P31m(e) 3.. P6 (g),(h)&03 (i)W) P321(c) ~& ? "7 9 f /~i } fh\— — D ( 1 ) ^ —f) XT O x^jtC//* I "L / 33 ty » v t* / 3 3 L/ P3 (c) ( S ) (P2z) c 4 6.. P6/mco(e) 3m. P63/mmo(e) PBjmcm(e) 3.m (h)\\Q, (i}\\0 P62e(e) P6322(e)

P3cl (c) P31c(e) (b ) (P2z) c 6 2.. P6222(e) P64 22(e)

.2-GElz 2 3m. P3ml (d) 3.. P321 (d) P3 (d)

( a )100/010/002. (b )100/010/003. ( C )010/110/002. ( d )100/010/001.

98 ( a ) (.2.GElz) 4 3.. P'Scl(d) c

P3xx 3 mm2 P6m2(j)> (k)00\ ..2 P312(j)j (k)00\

P3xx(z] 3 .777. P3ml (d)

( a ) (P3xx) 6 ..2 P6c2(j) c

P3x 3 m2m P62m(f),(g)00\ .2. P322(e),(f)00±

P3x[z] 3 . ,777 P31m(c) ( a ) (P3x) c 6 .2. P62c(g)

| 3 2 ..Pc+ Qlxx 3 ..2 2 3^ 6 ( d ) 3 r .PC~QIXX 3 ..2 P3~.12(a)00^ (b)OOl

1 3^. .PC+QIX 3 .2. P3 2 21(a)00l,(b)00i ( d ) 3 r .Pc-Qlx 3 .2. P3 1 21(a)OOjJ (b)OOl

G2z 4 3m. PS/rrmnCh) 3.. P62m(h) P622(h) P6/m(h) P31m(h) ( a ) (G2z) 2 8 3.. P6/moo(h) P6jmcm(h) o

E2z 4 3m. P6 ~/mmc (f) 3.. P62o(f)

P6 3/m(f) P31c(f)

R2z 6 3m. 3.. R32 (c) R3 (c) (°) '(R2z) c 12 3.. ff5c (c)

( a )100/010/002. (°)010/110/002. ( d )100/010/001.

99 N2z 6 2mm P6/mmm(i) 2.. P622(i) P6/m(i) ( a ) (N2z) c 12 2.. P6/mcc(i)

P6x 6 m2m P6/mmm(j) , (k)OO^ .2. P622(j), (k)00\ P3ml(g),(h)00l P6x[z] 6 . .m P6mm(d) ( a ) (P6x) c 12 .2. P6/moo(j)OOl P6 /mmc (i ) o

P6xx 6 mm2 P6/mrmi(l),(m)OOl ..2 P622(l) J> (m)00 12

L O J. til ( t* / • ( fl / \J\J 2 P6xxfz] 6 . m. P6mm(e) ( a ) (P6xx) 12 ..2 P6/moc(k)OOl c 3

..2P 3x 6 m2m P6 /mcm(g)00l c P6322(g)

. .2P 3x[z) 6 . .m c 3

.2.P E3xx 6 mm2 P6Jrmc(h)OOli c 6 ..2 PS 22(h)00~ P31c(h)00l .2.P E3xx [z] 6 .m. P63mo(C) c

3 1 .2PCc+ Qc lx 6 .2. P62 22(a) (d) V 2PCc~Q c lx 6 P6522(a)

3 l2 ' PCcEC\ 1XX 6 ..2 Z1™'™ (d) 3 2 2 - PCcVQ c lxX 6 ..2

Vz 6 2.. P6222(f)

(d) ~ Q2Z 6 f?SS * * P6422(f)

( a )100/010/002. ( d )100/010/00l.

100 c 6 .2. P6222(g),(h)00l (d) 3 r .Pc-Q2x 6 .2.

32 .. P Q2xx 6 ..2 ( d ) 3 . .P Q2xx 6 ..2 »;s:Z!

R3xx j [zj 9 .m. R3m (b)

R3x 9 .2. H32 (d),(e)00\

R6x 18 .2. a-trwan

.n. 1 R 3x 18 .2. RZc (e)OOl c

Hexagonal and rhombohedral, bivariant

P3xy 3 m. . P'S (j),(k)00l P3xy fz] 3 1 P3 (d)

3 2 ..P p^Qlxy fz] 3 1 P32 (a) ( d ) 3 1 ..Pc"Qlxy[z] 3 1 P3 2 (a)

P3xx2y 6 m. . P6m2(l) J (m)OOl P3xx2y fz] 6 2 P3m2(e)

P3xx2z 6 .m. P6m2(n)

. . 2P 3xy 6 171. .

,.2P3xy [z] 6 1 P3c2 (d)

P3x2z 6 . .m P62m(i)

P3x2y 6 m. * P3x2y [z 6 1 P31m(d) ( d )ioo/oio/ooi.

101 .2.P E3xy 6 777. . P62c(h)00l .2. P cE3xy[z] 6 1 P31c(c)

P6xy 6 777. . P6/m(j) 3 (k)00\ P6xy{z } 6 1 P6 (d)

2 r .P cE3xy 6 777. . .P E3xy{z] 6 1 P6 3 (c) 2 r c

P6xz 6 . .777 P'Zlm(k)

P6xxz 6 .777. P3ml(i)

•* -Qc lxy (z) 6 1 P6 2 (a) (d) Vi- 7Pp CcEF C yn Ixy I:slff 1 P6 5 (a)

32 ..P,-i Q2 Xy tz) 6 1 P6 2 (c) L - • f) 32xy{z] 6 1 P6 4 (o)

R3xy [z] 9 1 R3 (b)

P6x2z 12 . . m P6/mmm(n)

P6xx2z 12 .777. P6/mmm(o)

P6x2y 12 777. . P6/mmm(p)j (q)00\ P6x2y[ zj 12 1 P6mm(f)

.2.P 6xy 12 777. . P6/mcc(l)

.2. P 6xy{z] 12 1 P6oc(d) c

. .2P 3x2y 12 777. . "f^'001 . .2P 3x2y{z] 12 1

. .2P 3x2z 12 . .777 W/-.WOM c

()100/010/001.

102 .2.P E3xx2y 12 m. . P6^/rmc(j)OOi o .2.PcE3xx2y[z) 12 1 P6^nc(d)

.2.P E3xx2z 12 .m. P6-/rnma(k)OOl; c <6

R6xxz 18 . m. R3m (h)

R3xx2y fz) 18 1 R3m (c)

.n. T R 3xy [z) 18 1 R3c (b)

Hexagonal and rhombohedral, trivariant (site symmetry 1)

The 6-pointers: P3xy2z P% (I) .2.P E3xy2z P62c(i)00±

P3xx2yz P312(l) P6x2yz P622(n)

P3x2yz P321(g) I 3r 2pcc\lx2^ P6 2 22(c) P6 t.22(c) (d) 3 2' 2PCc" Q c lx2yZ o 3 2 ..Pc+ Qlxx2yz 2 ( d ) 3 . .P 'Qlxx2yz |^2 ..Pc+Q2x2yz P6 222(k) ( d ) 3 1 ..Pc-Q2x2yz P64 22(k) 3 . .P + Qlx2yz X.',™ ( d ) 3 1 ..Pc-Qlx2yz . .2P 3x2yz P6^22(i) o

P6xyz PS

The 12-pointers: 2 . .PcE3xy2z P6Jm(m2(o) P6xz2y P3lm(l) ..2P 3xy2z P6e2(l)00± . .cP E3xx2yz P31a(i)00i P3x2y2z P62m(l) ( d )100/010/001.

103 P6xxz2y P3ml(j)

The 18-pointers: R32 (f)

HZ (f)

The 24-pointers: P6x2y2z P6/rrmn(r) m. . P 6x2yz P6/mco(m)OOl c

P6jmem(l)00l . ,2P 3x2y2z o

P6 ry/mmo ( L ) 00 ij .2.P E3xx2y2z S

The 36-pointers: RSm (i)

.n. T R 3x2yz R3Q (f)OOl

104 Occurrence of Lattice Complexes Table 28

Lattice complex Multi Site Occurrence : in all its plicity symmetry space group, Wyckhoff letter, representations (oriented) shift from standard representation

Tetragonal, invariant p 1 4/mmm P4/mmm(a) J (b)OO^ (c)\\Q, (d)\\\ 42m P42m(a) J O>; Hi, (c)00\, (d)\\0 4m2 P4m2(a), (b)\\0, (c)\\\, (d)00\ 422 P422(a) , (b)00\> (c)\\0> (d)\\\ 4/m.. P4/m(a) J (b)OO^ (c)\\0, (d)\\\ 4. . P4 (a), Cb)00\, (c)\\0, (d)\^l Pfz) 1 4mm P4mm(a) s (~b)\\0 4.. P4 (a) 3 Cb)\\0 (a)pc 2 42m P4 2/mcm(a)OOl,(b)m 4m2 P4 2/mmc(e)OOl, (f)±H 422 P4/mcc(a)OOl, (c)\H 4/m.. P4/mcc(b),(d)\\0 4.. P42o(e), (f)\\0

M^W.'ft'iw' mmm. 777. mm P4 2/mam(b) ) (d)\ :kO 222. P42c(a)00l,(o)l\l

2.22 Sr^r^iu w^^^Ho'" 2/m. . ( a ) Pc fz2 2 4.. P4oo(a) 3 (b)\\0 2mm. P42mc(a), Cb)\\0 2. mm P42am(a) 3 (b)\\0 2.. P42 (a),(b)\\0 (b )c 2 42m P4/nbm(c)0$Of (d)0\\ 4m2 P4/nnm(a), (b)00\

( a )100/010/002. (b )110/110/001. From standard representation to cell.

105 422 P4/nbm(a), (b)00\ 4/m.. P4/mbm(a); Cb)00\ 4. . P42 1m(a)> (b)00\ P4b2(a),(b)00\ P4/n(a),(b)00\ mmm. P4/rrmn(e)0\\> (f)0\0 m. mm P4/nbm(o)0\\> (d)0\0 222. P42m(e) \OQ> (f) \0\

2.22 ™**(l)™o!?W\\ P42 2 2(a)> (b)OOl 2/m. . P4/m(e)0\03 (f)0\\ C(z) 4.. P4bm(a) 2mm. P4rm(c)\00 2. mm P4bm(b)lOO 2.. P4 (c)OlO ..2/m P4/nbm(e)^(f)^ ab

1 PP4^1^^ 42m I4/momCb)0\l 4m2 I4/mrm(d)Oll 422 I4/mam(a)OOl 4/m. . I4/mcm(c) 4.. P4/nno(d)\Ol P4/ncc(b) P42/nbc(d) P42/rribc(b)QOi P42/mnm(d)0\l P4p/ncm(b)

mmm. I4/mmn(c)0\0 m. mm I4/mcm(d)OlO 222. P4/moo(e)0\l P4/nnc(o)\00

(°)200/020/001. ( d )110/llO/002.

106 P4 2/nnm(c)0\0

I422(o)0\0 2.22 P4/mno(d)OH P4/nco(a)OOl P4 2/nbe(c)OlO

P42/rribo(d)0\l P42/nom(a)OOl I4c2(a)00l, (d)0\0 I422(d)0\l 2/m. . P4/mco(f)0\0 P4/rnno(o)0\0 P4 2/mom(f)OlO

P4 2/mnm(c)OlO

( d ) C c (z) 4 4.. I4cm(a) 2mm. I4mm(b)OlO 2. mm I4om(b)\00 2.. P42cm(a)0±0 P4 2nm(b)0\0 P4cc(e)0\0

P42bc(a) J (b)0±0 14 (b)0\0 J P2 8 ..2/m I4/mrn(f)lll

J P4/nnc(f)^l P4/noc(d)llO

( d )11071107002. ( e )200/020/002.

107

501-563 O - 73 - 8 0 2 4/mmm I4/mmm(a)j (b)00\ 42m P4 Jnnm (a) * (b) 00\ 2 I42m(a), (b)00\ 4m2 P4 2/nmc(a),(b)OOl I4m2(a)> (b)OO^ (c)0\l3 (d)0\l 422 P4/nnc(a),(b)OOl 1422 (a), (b)00\ 4/m.. P4/rmc(a) y (b)00\ I4/m(a),(b)00\ 4.. P42 1 c(a),(b)00\ P4n2(a), (b)00\ P4 2/n(a), (b)OOl 14 (a), (b)00\ 3 (o)0\l> (d)0\l mmm. P4 2/rmc(o)0\0> (d)0\\ m. mm P4 2/rmm(a), (b)00\ 222. P42o(b)lOl, (d)0\l P4 22(c)0\0; (d)0\\ 2.22 P4n2(o)0^(d)0^

2/m. . w^;oio!^oH Kz) 2 4mm I4mm(a) 4. . P4no(a) 14 (a) 2mm. P4 2mc(o)0\0 2 . mm P4 2nm(a)

2. . i^Tf-iP4 (lC-/V-yiLx n }n~n 2 ( b ) F 4 ..2/m P4 2/nrm (e)m,(f)m ^42^m"(dnu" J ( d ) F 8 4.. I4 2/aad(b) C 2.22 I4 1/aod(a)00l ( d ) F (z) 8 2.. c (e)I2 26 2 I4 l/acd(G)0 ^

)110/110/002. ( 6 )200/020/002.

108 D 8 42m F4 2/ddm2 ( f ) VD 4 4m2 I4 2/amd(a), (b)00\ 4.. I42d(a), (b)00\ I4 1/a(a),(b)00l 2.22 I4 2 22(a) y (b)OOl ( f ) VDEz) 4 2mm. I4 2md(a) 2. . I4 2 (a)

16 ..2/m F4 1/d^1 H( f ) VT 8 .2/m. I4 1/amd(o) ) (d)00{ 1 14^(0)^)00^

Tetragonal, univariant

P2z 2 4mm P4/mrm(g),(hmo 4. . P422(g),(h)l\0

2mm. P4m2(e), (f)\\0 2. mm P42m(g), (h)\\0 2. . P4 (e) 9 (f)l\0 ( a ) (P2z) c 4 4.. P4/moo(g) _, Ch)\\0 2mm. P4 9/mme(g) j (h) \\0 2. mm P4 2/mcm(g),(h)^0 2.. ™Z>'™™

(b ) C2z 4 4.. P4/nbm(g) P4/mbm(e) 2mm. P4/mm(i)OlO P4/nmm(f) 2. mm P4/nbm(h)0\0 P4/mbm(f)0\0

Unconventional setting. Cp. next entry. ( a ) 100/010/002. ( b )110/110 '001. ( f 2.. P42m(m)0\0 P42 2m(d) P4b2(e),(f)0\0 P422(i)0\0 P42 1 2(d) P4/m(i)0\0 P4/n(f) ( d ) (C2z) 8 4.. I4/mom(f) c 2mm* I4/mmm(g)0\0 2. mm I4/mem(g)0\0 2.. P4/mce(i)OlO P4/nno(g)\00 P4/mnc(f)OlO P4/ncc(e) P42/mcm(k)0\0 P4 2/nbc(f)0\0,(g) P4 2/nnm(h)0\0

P4 2/rmm(h)OlO P4 2/nom(f) I4e2(f),(g)OlO I42m(h)0\0 I422(f)0\0 I4/m(g)0\0

..2CIlz 2 4mm 4.. ^42^2(0)010 P4/n(o)0\0 2mm. P4m2(g)0\0 2. mm ~P4Zjm(e)0\0 2.. P4 (g)0\0 4 4.. P4/nec(c)0\0 c 2mm. P4 2/nmc(d)0\0 2» mm P4 2/nom(e)0\0 2..

)110/110/002. ( a )100/010/002.

no P4^9 2.2rd;oioj P4 ^/n(e)0\0

| PUxx [ tf 777. 2777 P4/rmrm(j ) 3 (k)OO^

..2 P4m2(h), (i)00\ P422(j),(k)00l

P4xx{z] 4 . .777 P4mm(d) ( a ) (P4xx) c 5 ..2 P4/mcc(j)OOl P4 2/mme(n)OOl ( ) CUx 5 .2. P4/nbm(k)> (l)00\ 7^ .2. I4/mam(j)OOl c

P4x 4 m2m. P4/mrnm(l), (m)OO^ M\\0, (o)\\\ .2. P42m(i), (j)lll, (k)00\, (l)\\0 P422(l), (m)\\\3 (n)OO^ (o)\\0

P4x[z} 4 .777. P4rm(e),(f)l\0

( a ) (P4x) c B .2. P4/rnoc(k)OOUl)lU^

( b ) C4xx 8 ..2 P4/nbm(i) J (j)00\ P4/nrrm(g), (h)00\ Cl \ ^ i C^4-xx) 16 . .2 I4/rmmCk)0\l c I4/mcm(i)OOl

.b.C2xx 4 777. 2777 P4/rribm(g)0\0,(h)Q\\

..2 P4b2(g)OlO,(h)0^

,b.C2xx{z) 4 . * 777 P4bm( o)0^0 ( a ) (,b.C2xx) B ..2 P4/rmc(g)0\l c

. .2P I2x 4 7772777. c P42/mmc(j) s (k)\\l, (l)OO^ (m)\\0 .2. P42c(g)00ij (h)\\l^ (i)\\l* (j)OOl P4g22(j) f (k)mt a)OOl,(m)HO

( a )100/010/002. ()110/110/001. ()110/110/002.

Ill ,.2P I2x(z) 4 .m. P4 2mc(d),MlO ( b ) .2.C F2xx 8 ..2 P4 2/nnm(k)OU,(l)0\l c P4 2/ncm(g)OOriJ (h)OOl

.2.P 2xx 4 m. 2m P4 9/mem(i), (j)00\ c ..2 P4o2(e)00^ (f)OOl P4 22 (n) 001, (o)OOl .2.P 2xx{z] 4 . .m P4 2om(d) ( b ) . ,2C 2x 8 .2. P4 e,/nbc(h)00^ (i)OOl c 2

.n. I2xx 4 m. 2m ..2 K&WH

.n.I2xx{z) 4 . .m P4 ^nm(c)

I2z 4 4mm I4/mmm (e ) 4. . P4/nno(e) P4/mno(e) 1422 (e) I4/m(e) 2mm. 2

I4m2(e) j (f)0\^ 2» mm P4Jnnm(g) P4»/mnm(e) I42m(e) 2.. P42c(m)0\l

L~X.ri,Li\&j+~P&Y1 9 ( & ) 1f '£KT /) D—L/ 2 —Lf P4 t>22(i)0\0

P4 2/n(f) 14 (e), (f)0\l ( d ) (F2z) 16 c 2.. I4 2/acd(d)

( b )110/110/001. ( d )110/110/002.

112 4...P I Ix 4 .2. 3 cc c (g) H,..P I Ix 4 .2. 1 cc c Z$™wal

H...P VDlxx 4 3 cc ..2 (g) 4,..P VDlxx 4 ..2 P4 1 22(c)00\ 1 cc

v 4 . .2 (g) 4,.. I vDlxx 4 1 c iy

I4XX 8 777. #777 I4/mrm (h ) ..2 P4/nnc(k) P4 2/nmc(f) I4m2(g), (h)O^ 1422 (g) 8 . .77? I4mm(c)

I4x 8 I4/mmm (i) ^ (j)\ \0 .2. P4/nnc(i),(j)OOl P4r>/nnm(i)s (j)OO^ I42m(f),(g)00\ 1422 (h), (i)00\ 8 . 777. I4mm(d)

.b.C 2xx c 8 777. #777 I4/mcm(h)0\0 ..2 P4/ncc(f)OOl

1422 (j)OU

.b.C 2xx{zL 8 . ,777 c

VD2z 8 2mm. I4 2/amd(e) 2.. I42d(e) I4 2 22(o) I4 2/a(e)

5.. VTF Ix 8 .2. c I42d(d) ( g )100/010/001.

113 ..dVD2x [z) 8 . m. I4 2md(b)

.2. VD2xx 8 ..2 I4 2 22(d) ( g ) ,2. VD2xx 8 ..2 I4 2 22(e)

..22 VTC Ix 8 .2. I4 2 22(f) cc

..2 VT2x 26 .2. I4 1/amd(f) v D4xx 26 ..2 14 /amd(g)

U..I 2Pc2 lx 26 .2. I4 1/aed(e)OH

.2.F 2xx 26 ..2 I4 1/acd(f)00l c

Tetragonal, bivariant

PUxxz 4 . . m P42m(n)

•2 1 .CIlz2xx 4 . .m P42 1m(e)0\0

P4xz 4 .777. P4m2(j),(k)l\0

P4xy 4 m. . P4/m(j),Ck)OOl P^xy(z) 4 1 P4 (d)

5 1 . .P I2xy 4 m. . P4 2/m(j) 5 1 . .P I2xy[zj 4 1 P4 2 (d)

. 3 ..PccVDI c lxy|(z} 4 1 P4^ (a) (g) V- pcc VDIclxy[zl 4 1 P4 2 (a)

P4x2y 8 m. . P4/rrmn(p), (q)00\ PUx2y{z) 8 1 P4mm(g)

( g )100/010/001.

114 PUxx2z 8 ..m* P4/mmm(r)

P4x2z 8 .m. P4/mmm(s)j (t)\\0

.2.P 4xy 8 m. . P4/mcc(m) .2.P 4xy{z) 8 1 P4oc(d) c

. . 2C4xxz 8 . . m P4/nbm(m)0\0

..mC^xy 8 P4/mbm(i)> (j)00\ . .mC4xy[z) 8 1 P4bm(d)

,b.C2xx2z 8 . .m P4/mbm(k)0\0

. . 2I4xy 8 P4/rmc(h) ..2I4xy[z) 8 1 P4nc(c)

. .mC4xz 8 .m. P4/nmn(i)

. . 2CIlz4xx 8 . .m P4/*nmm (j ) 0 2 0 m. .P Uxz 8 .m. P4 f./mmc ( o ) 00 u _, (p / ^ 2 u c

..2P I2x2y 8 m. , P4 /mmc(q) c . .2P I2x2y[z] 8 1

.2.P 2xx2y 8 m. . P4 /mom(n) .2.P 2xx2y [z} 8 1 m. . P Hxxz 8 . .m P4 ~/mem (o)00\ c 2

8 . .m P4 Jnnm(m) ..2I4xxz ci

.b2C c2xy 8 m. . P4 2/rrbo(h) .b2Cc2xy{z) 8 1

.n.I2xx2y 8 m. . ,n.I2xx2y{z) 8 2 P42nm(d)

115 .n. I2xx2z 5 . .7* P4 2/mrm (i)

| 5 . 777. P4 /nmc(g)

5. .F2xxz 5 . .77? P4 p/nornd) J|^

IHxz % 5 . 777. I2m2fi; •

I4XXZ 5 ..m i32™ft:;

I4xy 5 777. . I4/m(h) I4xy[;s) 8 1 14 (c)

4 . . VD2x}t fz] 8 1 14 (b)

IHx2y 16 m. . I4/mmm(l) I4x2y[z) 16 1 14mm (e)

I4xx2z 16 . .777 14/rrmn (m)

I4x2z 16 .777. I4/mrrm(n)

. .mC Hxy 16 777. . 14 /mom (k)

. .mC 4xy{z] 16 1 I4om(d)

• b . C M- xxz 16 . .77? 14 / mom CL) 0~2^ c

. 2 . VD4xz 16 .m. I4 2/amd(h)

..dVD2x2y (z) 16 1 I4 2md(c)

.bdF 2xy (z} 16 1 I4?d

116 Tetragonal, trivariant (site symmetry 1)

The 4-pointers: 1 P4xyz P4 (h) PUxy2z P4/m(l)

The 8-pointers; m. .P Uxyz P42/m(k)00^

IC^xyz P4/n (g)

n. .I4xyz P4 2/n(g)

iHxyz 14 (g)

The" 16-pointers : P4x2y2z P4/mrrm (u)

m. .P 4x2yz P4/mcc(n)OOl

. ,mCHx2yz P4/nbm(n)

. .cIHx2yz P4/nno(k)

. .mC4xy2z P4/rrbm(l)

. .2I^xy2z P4/rnne(i)

. .mCM-xz2y P4/nnm(k)

. ,c2C 4xy2r P4/ncc (g)

m. .P Uxz2}r P42/mmc(r)00 :k

m. .P 4xxz2y P4 2/mcm(p)OOi

.22C 4xyz P4jnbo(k) * ( g )100/010/001.

117 . .2I4xxz2y P4 2/nnm(n) m.2C 4xyz P42/rrbc(i)OOl

.n.I2xx2y2z P4 /rmm(k)

. . cI4xz2y P4 /nmc(h)

. .m2C 4xyz P42/ncm(j)

I4xz2y I4m2 (j)

. .2C 4xyz I4c2(i)

I4xxz2y I42m(j)

,2. VD4xyz I42d(e)

I4x2yz 1422 (k)

.2. VD2xx2yz I4 2 22(g)

I4xy2z I4/m(i) a. . D4xyz I4 2/a(f)

The 32-pointers: I4x2y2z I4/mrm(o)

.b.C 4x2yz I4/mcm(m)OOl

.2. VD4xz2y I4 2/amd(i)

.22F 4xyz I4 2/acd(g)

118 Occurrence of Lattice Complexes Table 29 Lattice Transfor Multi Site Occurrence: complex mation from plicity symmetry space group a Wyckoff letter. in all its standard (oriented) shift from standard representation represen- i representa tations tion to cell

Orthorhombic, invariant

0 1 mmm p-.f«;.ftJiw1 woM, iWW}. ii 222 P222(a), (b)lOO, (o)0\0, (d)00\,

P{z) 1 mm2 Pmm2(a), (b)0\0y (o)\00> (d)\\0 P 200/010/001 2 .2/m. Pmma(a), (b)0\0, (c)OO^ a (d)0\\ P fz) 200/010/001 2 ..2 Pma2(a),(b)0\0 ! a P 100/010/002 2 222 Pcom(e)00ls (f)lOl> (g)OH^ (ln)\\\ c ..2/m Pecm(a)> (b)l\0, (o)0\03 (d)\00 P Izl 100/010/002 2 ..2 Pce2(a),(b)0\0,(e)\00,(d)\\0 c Pab 200/020/001 4 222 Cmma(a)lOO, (b)lO\ 2/m.. Cmma(c), (d)00\ .2/m. Crrma(e)HO>(f)m ..2/m CTrmm(e)^0,(f)M\ 1 Pban(e)^(f)m

Pab ( z} 200/020/001 4 ..2 Cmm2(c)llO

P ac 200/010/002 4 1 Peca(a),(b)0\0 Pbc 100/020/002 4 1 Pbcm(a),(b)lOO Pbc (z} 100/020/002 4 ..2 Abm2(a)>(b)\00 ?2 200/020/002 8 222 Fmmm ( f ) ij u 5 2/m. . Pnwvft (c ) 0 u ij .2/m. Fmrm(d)lOl ..2/m Fmm(e)V*0 1 Cmoa ( o ) ij nO Ccoct ( o) i^O u » (d)0~^"fi

119 Immm(k) JJ J

P2 U) 200/020/»2 8 ... Fmm2(b)UO 0 2 mmm Cmmm(a), (b)\00, (c)\Q\, (d)00\ 222 Pban(a),(b)lOO,(c)m,(d)00± C222(a), (b)0±0, (c)\0\, (d)00\ ..2/m Pbam(a), (b)OO^ (c)0\Q, (d)O^ Cfz) 2 mm2 Cmm2(a), (b)0\0 ..2 Pba2(a),(b)0\0 Alz) 001/100/010 2 mm2 Amm2(a) j (b)\00 ..2 Pno2(a) 3 (b)\00 B 010/001/100 2 2/m.. Prma(a), (b)\00, (c)\\0, (d)0\0 C 100/010/002 4 222 °^a^b°^ c

2/m.. Cmom(a), (b)0\0 ..2/m Cccm(c), (d)0\0 Ibam(c), (d)lOO 1 Pbcn(a), (b)0\0 C (z) 100/010/002 4 ..2 Ccc2(a), (b)0\0 C Iba2(a), (b)0\0 A 002/100/010 4 .2/m. a Irma(c)ill,(d)Hl 1 Pnna(a),(b)00\ A a {zj 002/100/010 4 ..2 Ama2(a) Ima2(a) Bb 010/002/100 4 1 Pnma(a), (b)00\ 2/m.. Inma(a) 9 (b)00l

0 2 mmm Immm(a), (b)0\\, (c)\\0, (d)\0\ 222 Pnnn(a), (b)\00, (c)00\, (d)0\0 I222(a), (b)lOO, (c)00\, (d)0\0 ..2/m Pnnm(a), (b)OO^ (c)0\Q, (d)0\\ I[z) mm2 Imm2(a), (b)0\0 ..2 Pnn2(a), (b)0\0

120 0 4 ffin?n Fmmm(a) „ (b)00\ 222 Ccca(a), (b)00± F222(a), (b)00l3 (o)ll^ (d)ltt 2/m.. Cmca(a), (b)00\

..2/77? Cocm(e)UO, (f)UO

1 Pnnn(e)Hl, (f)lll Pccn(a), (b)00\ Pbca(a), (b)00\ F(z) 4 mm2 Fmm2(a) ..2 Ccc2(c)UO Aba2(a)

0 8 222 Fddd(a), (b)00± DfzJ 8 ..2 Fdd2(a)

0 16 1 ?ddd(e)>(d)\\\

Orthorhombic, univariant

P2x 2 2mm Pmmm(i), (j)OO^ (k)0\0, (l)0\\ 5. . P222(i), (j)OO^ (k)OlO, (l)0\\ P2x{z} 2 .m. Pmm2(e), (f)0\0 P2y 001/100/010 2 m2m Prrmn(m), (n)00\, (o)\003 (p)\0\ .2. P222(m) > (n)00\ y (o)\OQ3 (p)\0\ P2y[z] 001/100/010 2 m. . Pmm2(g) 3 (hJ^OO P2z 010/001/100 2 mm2 Pmmm(q), M0\0> (s)\00, (t)\\0 ..2 P222(q), (r)\00, (s)0\0, (t)\\0 (P2x) c 100/010/002 4 2. . Pccm(i)00l,(j)0±i (P2y) a 002/100/010 4 .2. Pmma(g), (h)00\ (P2y) c 001/100/020 4 .2. Pocm(k)00l, (l)\0l (P2z) c 010/001/200 4 ..2 Pccm(m), (n)\\03 (o)0\0, (p)\00 (P2x) ab 200/020/001 8 2.. Cmmoi ( h ) ^ ('L)OO^ (P2y) ab 002/200/010 8 .2. CmfncL ( j ) nOO j (k) ~$0\ (P2z) ab 020/002/100 8 ..2 Cmnundn) i^O Cmma(l)lOO

121 (P2x) 2 200/020/002 16 2.. FmmdHtt (P2y) 2 002/200/020 16 .2. Fmmm(k) jJi (P2z) 2 020/002/200 16 ..2 Fmmm ( j ) ^ ij 4

.2.P Biz 2 Pmmo. (&) ^00 j ( f) ^^O a .2.P Blx 001/010/100 2 2.. 2..P cAly 010/001/100 2 .2. P222 2 (c)00^(d)\0l 2..P Aly(z) 010/001/100 2 m. . Pme2 (a), (b)\00 ..2P Clyfz) 100/001/010 2 m. . Pma2(o)^00 a (.2.P Biz) 100/010/002 4 ..2 Pcca(d)lOO>(e)l\0 a c (..2PClx) 001/100/020 4 2.. Pbcm(c)OlO b c (2..P Aly) 020/001/100 4 .2. Prma(g)OlO c a (2..P Aly) , 020/002/100 8 .2. Cmca(e)\ll c J ah

2 1 ..CIlz 2 mm2 Pmrnn(a), (b)0\0 ..2 .. 2lBIly 010/001/100 2 m. « 100/010/002 4 ..2 1* ' c Pccn(e) J J^ (d) 110

C2x 4 2mm Crrrnn(g)^ (h)00\ 2.. Pban(g),(h)00\ C222(e),(f)00\ C2x(z) 4 .m. Cmm2 (d) C2y 010/100/001 4 m2m Cnrnn(t), (j)00^2 .2. Pban(i),(j)00\ C222(g),(h)00\ C2y(z) oJo/ioo/ooi 4 m. . , Cmm2(e) B2x 100/001/010 4 2.. Pmna(e),(f)0\0 A2y[z) 001/100/010 4 m. . Amn2(d),(e)±00 (C2x) 100/010/002 8 2.. Cmcm (e) C Cccm(g)00l Ibam(f)00l (C2y) c 010/100/002 8 .2. Cccm(h)00l Ibam(g)00l (B2x)b 100/002/010 8 2.. Irrma(f) (A2y) 002/100/010 8 .2. Inma(g)lll Cl

122 C2z 4 rrm2 Crmm(k)^ (l)0\0 ..2 Pban(k), (1)0^0 Pbam(e), (f)0\0 C222(i),> (j)OlO A2x(zl 001/100/010 4 .m. Amm2(o) (C2z) 100/010/002 8 ..2 CoQm(i)* (j)0\0 c Ibam(h), (i)0\0

2 r .CcFly 4 m2m .2. Pbcn(c)00l

2 1 ..C cFly[z) 4 m. . Cmc2 (a) .2-.C Fix 4 2.. C222 2 (a) 1 c oTo/ 100/001 . .2.B, Fix 010/001/100 4 2.. Bnna(d)m 1 b ..2,A Fly{z) 001/010/100 4 m. . Ama2(b)±00 _L a

2. .P Flz 4 mm2 Cmma(g)OlO ab ..2 C222(k) HO ..2P Fly 010/001/100 4 .2. Pcca(c)00l ac J . .2P, Flx(z) 001/010/100 4 .m. Abm2(c)OlO be (2..P .Flz) 100/010/002 8 ..2 Ccca(h)HO ab c Iboa(e)OlO (.2. P. Fix) 002/100/010 8 2.. Ibcat c)00l be a ( . . 2P Fly), 010/002/100 8 .2. Ibca(d)lOO

I2x | 4 2mm Jmmm(e)^^ /i 3 (jJO-^U/^*i/oi/~i 2.. Pnnn(g), (h)00\ 1222 (e), (f)00\ I2x{zJ 4 .m. Imm2(c) I2y 001/100/010 4 m2m Immm(g), (h)00\ .2. Pnnn(i)> (j)lOO I222(g) 3 (h)\00 I2ytz} 001/100/010 4 m. . Imm2 (d) I2z 010/001/100 4 rrm2 Irrmntt), (j)\00 ..2 Pnnn(k), (l)0\0 Pnnm(e), (f)0\0 I222(i),(j)OlO

123

SOI-563 O- 73 - S ,2-B^A Iz 4 mm2 b a Irrma(e)OlO ..2

. . 2C Bv Ix 001/100/010 4 c b 2. .A C ly 010/001/100 a c y 4 .2. I2 2 22 2 2 (b)^0 2..A C ly(z} 010/001/100 4 .2. Ima2(b)lOO a. C

F2x 8 2mm 2.. Cmoa (d) Ccoa(e)

F2x(zl 8 .m. Frm2(d) F2y 001/100/010 8 m2m Fmnm(h) .2. Ccca(f)

F2y[z) 001/100/010 m. . Fmm2(c) F2z 010/001/100 mm2 Fmmm(i) ..2 Coom(k)^0 Coca (g) F222(g) _, (h)lH

D2x 16 2.. Fddd(e) D2y 001/100/010 16 .2. Fddd(f) D2z 010/001/100 16 ..2 Fddd(g)

Orthorhon±>ic, bivariant

P2y2z 4 m. . Pmmm(u); (v)\00 P2x2z 001/100/010 4 .m. Prrmm(w) 9 (x)0\0 P2x2y 010/001/100 4 . .m Prmm(y), (z)00\ P2x2ytzJ 010/001/100 4 1 Pmm2(i)

2..Pc2xy 4 . . m Poom(q) 2..Pc2xy{zJ 4 1 Pcc2(e)

124 m. .P 2xz 4 .m. Prrmatt), (j)0±0 a m. .P 2xy{z) 100/001/010 4 1 Pma2 (d) a

.2.P Blz2y 4 m. . Pmma(k)^00 a , , 2..P Aly2x(zJ 010/001/100 4 1 Pmo2 (c)

. 2 . B2yz 4 m. . Prnna(h) 2..A2xy{z) 010/001/100 4 1 Pna2(o) b..C2xy 4 . .m Pbam(g),(h)00\ b..C2xy{zl 4 1 Pba2(e)

2 - 1PbcVcFlx^ 4 . .m Pbcm(d)00i • 2lPacBaC cFlxy (zl 010/100/001 4 1 Pea21 (a) n. ,I2xy 4 . .777 Pnnm(g)

n. ,I2xy{zl 4 1 Pnn2(o)

2 1 ..CIlz2y 4 m. . Pmmn(e) -2 1 .CIlz2x 010/100/001 4 .777, Pmmn(f)

..2 1BIly2x[zJ 010/001/100 4 1 Prm2 1 (b)

1.2- B. A FI Ixz 4 .777. Pnma(o)OlO 1 b a a "12 r CcAaFI a lxy Iz} 100/001/010 4 1 Pna22 (a)

C2y2z 8 777. . Cmmm(n) C2x2z 010/100/001 8 .777. Cmnvn(o)

A2x2y{zJ 001/010/100 8 1 Amm2(f)

C2x2y 8 . • 777 CTrmn(p), (q)OO^

C2x2y{zJ 8 1 Cmm2 (f)

.n.C 2yz 8 777. . Cmcm(f)

. .nA 2xy|z} 001/010/100 8 1 Ama2(c) a

2 l" CcFly2x 8 . . 777 Cmcm(g)00l

2 r .C cFly2x(z] 8 1 Cmo2 (b)

125 .2.F2yz 8 m. . Cmoa (f) .2.F2xy(z} 001/010/100 8 1 Aba2(b) n. .C F2xy 8 . .m Cccm(l) n..C F2xy(z) 8 1 Ccc2 (d) c

.m.P ab 2yzJ 8 m. . Cmma (m) m. .P V 2xz 010/100/001 8 .m. Cmma(n)0±0 ab .m.Pbc 2xy(z} 001/010/100 8 1 Abm2 (d)

I2y2z 8 m. . Imrnm (I) I2x2z 001/100/010 8 .m. Immm(fn) I2x2y 010/001/100 8 . .m Immm(n) I2x2yfz) 010/001/100 8 1 Imm2 (e ) b. .C 2xy 8 . ,m Ibam(j) b. .C 2xy(z] 8 1 Iba2(c)

.2.Bb 2yz 8 m. . ImmaCk) 2. .A 2xz 010/100/001 8 .m. Imma(i) iH a 2. .A 2xy(z] 010/001/100 8 1 Ima2(c) a

F2y2z 16 m. . Frrmn(m) F2x2z 001/100/010 16 .777. Fmmm(n) F2x2y 010/001/100 16 . . m Fmmm(o) F2x2y[z) 010/001/100 16 1 Fmm2(e)

d..D2xy {zl 16 1 Fdd2(b)

126 Orthorhombic, trivariant (site symmetry 1)

The ^-pointers: b2.C 2xyz Pbon(d)

bc.F2xyz Pbca(c)

.maB 2xyz Pnma (d)

C2x2yz 0222(1)

.2 .C Flx2yz C222 2 (c)

I2x2yz 1222 (k)

. .2C B,lx2yz I2 1 2 1 2 1 (d)^0^ c b J

The 16-pointers:

127 F2x2yz F222(k)

The 32-pointers: F2x2y2z Fmmm(p) d. ,D2x2yz Fddd(h)

128 Occurrence of Lattice Complexes Table 30 Lattice complex Multi Site Occurrence : in all its plicity symmetry space group, Wyckoff letter, representations (oriented) shift from standard representation

Monoclinic, invariant 0 1 2/m P2/m(a), (b)0\0, (c)00\, (d)\00, P{xz} 1 m Pm (a), (b)0\0 P[y> 1 2 P2 (a), (b)OOl, (c) \00, (d) \0\ (3) ph 2 1 P2 /m(a), (b)\003 (e)00\, (d)\0\ 2 1 P27c(a), (b)\\03 (c)0\09 (d)\00 ( )Pab 4 1 C2/m(e)W3 (f)m

00 2 2/m C2/m(a), (b)0\0, (c)00\, (d)0\\ CfxzJ 2 m Cm (a) C{y] 2 2 C2 (a), (b)00\ ( d )A 2 1 P2 1/c(a), (b)lOO, (c)OO^ (d)\0\

( b ) C c 4 1 C2/c(a), (b)0\0 ( 6 ) F 4 1 C2/c(c)W,(d)lU

Monoclinic, univariant

P2y 2 2 P2/m(i), (j)lOO, (k)OO^ (l)\0\ P2y{xz] 2 1 Pm (c)

CP c Alyy 2 2 P2/c(e)00^(f)±Oi CP cAly{xz) 2 1 PC (a)

C2y 4 2 C2/m(g),(h)00\ C2y{xzJ 4 1 Cm (b)

( a )100/020/001. ( b )100/010/002. (°)200/020/001. ( d )001/010/100. ) 100/010/102. From standard representation to cell.

129 1C Fly 4 C2/c(e)OOl 1C Flyfxz] 4 Co (a)

Monoclinic, bivariant

P2xz 2 m P2/m(m), (n)0\0 P2xz[y] 2 1 P2 (e)

2 R ACIlxz 2 m 1 b 2. .. r AC 1 Ixz I y J 2 1 P2 1 (a) 1 Jb

C2xz 4 m C2/m(i) C2xz[y] 4 1 C2 (o)

Monocllnic, trivariant

P2xz2y 4 1 P2/m(o) mP 2xyz 4 1 P2 1/m(f)

2P 2xyz 4 2 P2/c(g)

cA2xyz 4 1 P2 1/o(e)

C2xz2y 8 1 C2/m(o)

2 C F2xyz 8 1 C2/e(f)

130 Occurrence of Lattice Complexes Table 31 Lattice complex Multi Site Occurrence : in all its plicity symmetry space group, Wyckoff letter, representations (oriented) shift from standard representation

Anorthic (triclinic), invariant p 1 1 PI (a) t (b)OOlt (c)0&f (d)iOO,

Plxyz) 1 1 PI (a)

Anorthic (triclinic), trivariant

P2xyz 2 1 PI (i)

131

221 221

223 223

225 225

-228 -228

-229 -229

y

8

h

n

h

u

L

°h"

0,5-222

0^-224 0^-224

°h~

Ou

0^-226 0^-226

oJ-227 oJ-227

O

0

I6z4x2y I6z4x2y

48(1)1

..2I6z2xx2y ..2I6z2xx2y

48(n)l

P6z4x2y P6z4x2y

6z4xx 6z4xx

L

? ?

]

m

. .

6z4xy 6z4xy

.

)1

^ ^

9

.P

192(1)]

F6z4x2} F6z4x2}

. .

48(1 48(1

1

m

..2I6z2x2y ..2I6z2x2y

.2I6z2xx .2I6z2xx

24(m) 24(m)

P6z4xx

24(k)..m

. .

.m

. .

I6z4x

-U

.

(k) (k)

.

F6z4x F6z4x

)m. )m.

96 96

6xx

2I6z2x 2I6z2x

6z2x 6z2x

2

(1 (1

l(k)m. l(k)m.

2

-F

2^

(h)l

6z2xy 6z2xy

.2P

4- 4-

2

96(i)m. 96(i)m.

3

. .

kinds) kinds)

i i

192 192

1 1

2l

3 3

I6z4xy I6z4xy

i

14

. .

48(1)1

F6z4x F6z4x

d.

. .

24(k)

P6z4x P6z4x

96(j)m..

i i

of of

i i

,3.W4xx ,3.W4xx

24(j)..2

192(1)1

,.2D6z2xx2y ,.2D6z2xx2y

.2

24(i)

. .

3. 3.

96(h)..2

£^4..F6xx £^4..F6xx

I12xx

(j)m.m2

(h) (h)

(i)m.m2

I12xx I12xx

ISxxx ISxxx

96(g)..2

directions, directions,

16(1). 16(1).

. .

4..T6xx 4..T6xx

96(h)..2 96(h)..2

12(1) 12(1)

48(h) 48(h)

,3.J*4x ,3.J*4x

24(h)2.. 24(h)2..

J*4x J*4x

. .

< <

n2.. n2..

.W2z .W2z

64(g),3. 64(g),3.

2 2

,3.J*4x ,3.J*4x

.m

.3

24(g)2. 24(g)2.

.mm

symmetry symmetry

. .

mm

(h)mi

0,

2 2

iH-3 iH-3

J4x J4x

2. 2.

(I6z)

- -

96(f)2

(13 (13

.2D6z2x: .2D6z2x:

3. 3.

. .

(g) (g)

.3. .3.

. .

I6z I6z

96(g) 96(g)

12(h)mm2.

48(g) 48(g)

12 12

m3m m3m

48(f)4.. 48(f)4..

ISxxx ISxxx

2 2

(g) (g)

ISxxx ISxxx

.3.W2z .3.W2z

16(f) 16(f)

3m

. .

3m

mm

. .

.

System System

2 2

P8xxx P8xxx

W* W*

8(g) 8(g)

FSxxx FSxxx

D6z D6z

..2I4xxx)

32(f) 32(f)

I6z I6z

64(e).3. 64(e).3.

(P6z)

12(f)

Cubic Cubic

48(f)2.

I6z

z z

6z 6z

12(f)2.22 12(f)2.22

e)4..

m m

(

I I

48(e)mm2.

m

12 12

2

32. 32.

3m

4m. 4m.

. .

.3m

HJP6 HJP6

(f)4m.

F6z F6z

(e) (e)

*Hp

(e) (e)

8{e).32

.2I4xxx .2I4xxx

.2D4xxx

8(e) 8(e)

24 24

W* W*

Z Z

W*

. .

Table Table

32 32

48(d)4..

J

(e) (e)

2 2

2(d)4 2(d)4

6z 6z

1

6

P

w

\ \

24(d)4/m.. 24(d)4/m..

4m.

T

H H

2

F

ip ip

55 55

(d) (d)

3 3

J J

p p

J* J*

8 8

p

2 2

; ;

3 3

2 2

8 8

1 1

(d).3m

: :

3 3

J

I I

1 1

2 2

24(d)m.mm

6(d)42.m

1 1

32(c).3.

(c) (c)

8(c).3. 8(c).3.

w w

5C5

(d)4/mm.m (d)4/mm.m

6 6

2

24(c)4m.2 24(c)4m.2

T T

. .

i5sF i5sF

j j

16(c) 16(c)

i«P

(c).3m (c).3m

F F

3(c) 3(c)

* *

42.2 42.2

8(c)43m 8(c)43m

* *

1 1

imm. imm.

J J

1 1

J

2 2

1 1

888*2 888*2

P

32(b).32 32(b).32

D

2 2

6(b) 6(b)

JucF

2 2

8(b)m3. 8(b)m3.

4(b)

6(b)ir 6(b)ir

UJP UJP

5 5

(b)43m

JHF JHF

(b)m3m

2 2

I I

1 1

T T

P

I I

D D

P P

F F

l(a) l(a)

2{a)m3. 2{a)m3.

2(a)432 2(a)432

8(a) 8(a)

8(a)432 8(a)432

4(a) 4(a)

2(a)43m 2(a)43m

16(a)23. 16(a)23.

cH

Pm3m

Fd3m

Fm3m

Fm3c

Pn3m

Im3m

Pm3n

Pn3n

Ia3d Fd3c

-217 -217

d

T^-219 T^-219

T T

TJ-215 TJ-215

24(j)l

P6z2xx2y P6z2xx2y

kinds)

3 3

.'.m

of of

P6z2xx P6z2xx

12(i) 12(i)

directions, directions,

,3.J4x ,3.J4x

12(h)2..

,

T T

- -

symmetry symmetry

43m 43m

(13 (13

(g)2.mm

iHP^z iHP^z

48(f)

System System

6(f) 6(f)

P6z P6z

Cubic Cubic

32. 32.

32(e).3.

Table Table

(d)?..

-*SJF -*SJF

(d)43m

24(c)

3(c)

(c) (c)

JJJF JJJF

J*

6(b)?2.m

iP

2 2

(b)23.

(b)

(b)4"3m

I

I I

2 2

P

F

8(a)

2(a)23.

2(a)4"3m 2(a)4"3m

Ha)

4(a)

I43d

F43c

P43n P43n

I43m I43m

F43m F43m P43m. P43m. Table 32. Cubic System (13 symmetry directions, of 3 kinds)

432 - 0

0 -207 P432 HsP J sHj P6z PSxxx ,3.J4x P12xx P6z4xy

(b)432 3(c) (d)42.2 6(e) .. 8(g).3. 12(h)2.. 12(i) 24(k)l

P4 2 32 I C55F 5 HP J* W sHw ..2I4xxx I6z .3.W2z IH.3.W2z c's4 2 ..F3xx 0 -208

2(a)23. 4(b) (c).32 6(d}222.. 6(e) (f)2.22 8(g).3. 12(h}2.. 12(i) (D..2

F432 F ?HF U^P 2 J 2 F6z F8xxx Fl2xx 555 0 -209

4(a) (b)432 8(c)23. 24(d)2.22 24(e}4.. 32(f).3. 48(g) 48(i)2.

D 1HD T HsT ,.2D4xxx D6z 22..T3xx . .2D6z2xy 0 -210

8(a)

I J* iUp W* I6z I8xxx .3.J*4x I12xx I6z4xy 0 5-211

2(a)432 6{b)42.2 8(c).32 12(d)2.22 12{e}4.. 16(f}.3. 24(g)2.. 24(h)..2 24(i) ..2

0 -212 (b).32

0 ?-213

~V 22,.Y*2xxx ,3.V2z 4 3 .. Y*3xx 22. .Y*3xx2yz 08-214

(b).32 12(c) {d)2.22 16(e).3. 24(f)2.. 24 (h) 4 -20!

-202

-204

'-200

2

3

5

h

T

T/-206

1

)

1

(

P6z2x2y 24

(k)m..

H?P6z2x

96(i)l

kinds)

F6z2x2y

3

P6z2x 12(j)

of

48(h)l

I6z2x2y

F6z2x .

48(h)m..

.I6z2xy

PSxxx

24{h)l

n.

directions,

I6z2x

24(g)m.

T,

48(g)2..

-

symmetry

,3.J*2x

12(g)2..

m3

ISxxx

16(f).3.

(13

96{g)l

d..D6z2xy

FSxxx

32(f).3.

I6z

System

12{f)2..

6xyz

3.J*2x

.

P

D6z

12(e)mm2..

48(f)2..

Cubic

48(e)l

F6z

(h)mm2.. 22..

24(e)mm2..

32.

..2I4xxx

8(e).3.

I6z

P6z

6(e)

..2D4xxx

32{e).3.

Table

12{d)mm2..

,F6xyz

24(d)2..

J*

J-

24(d)l

be. 2

6(d)222.. 24(d)2/m..

HIT

ttl* c).3.

(d)mmm..

8{c).3

2xxx

2

,F2xxx

T

8(c}.3.

16{c).3.

3(c)

16(c)

8(c)23.

be.

22..P

D

4(b)

.1.

6(b)mmm..

j

5

HiF

HlF 5

(b)m3.

(b)23.

(b)

(b).3.

F

(a)

D

2{a)23.

4(a)

8(a) 2(a)m3.

4

8(a)

Pm3

Pn3

Fm3

Fd3

Im3

Pa3

Ia3 -1

-196

-197

-198

-199

5

T

kinds)

3

6(g)

.3.J2x

of

directions,

symmetry

P6z 6(f)

23-T

F6z

(13

24(f)

,3.J*2x

12(e}2..

System

P4xxx

4(e}.3.

I6z

Cubic

F4xxx

16(e).3.

Y*lxxx3yz

12{d)2..

2

,P

32.

24

.

T

J

l

2

i

2

1

l

2

1

2

(d)222..

Table

(d)23.

I4xxx

8{c).3.

J

3(c)

..FYlxxx3yz

(c)

,

J*

6(b)222..

(b)

(b)23.

P

F

Ka)

4(a)

2(a)23.

3

1

P23

F23

123

P2

12,3

c c

' '

(k)m2m

description

' '

(P6xx)

(N2z)

12(i)..2

(P6x),

6(j)

P6x P6x

c c

kinds) kinds)

3 3

hexagonal hexagonal

(G2z)

8(h)3.. 8(h)3..

2mm

(G2z)

8(h)3..

in in

(i) (i)

N2z N2z

6 6

kinds) kinds)

2 2

6(f)222

directions,of directions,of

G2z

4(h)3m.

,2/m,

6(g) 6(g)

symmetry symmetry

c c

mmm

N

(7 (7

.

6h 6h

OO^N

directions,of directions,of

(g) (g)

D

6(g)2/m.

)1

E3xx2y2z E3xx2y2z

6(f)..2/m

- -

c

(q)m. (q)m.

24(1 24(1

OO^P6x2y OO^P6x2y

System System

N N

3(f)

c c

symmetry symmetry

6/mmm 6/mmm

00^.2.P

(P2z),

(4 (4

4(e)6.«

(P2z)

)1

3x2y2z 3x2y2z

4(e)3.m 4(e)3.m

c

Hexagonal Hexagonal

(P2z)

24(1 24(1

(e)6mm

System System

P2z P2z

q..2P

2 2

33. 33.

6x2yz 6x2yz

c

4(c)3.2 4(c)3.2

(c

.P

E3xx2z E3xx2z

24(m)l

12 12

P6x>

(k).m.

(d)?m2

^m. ^m.

Table Table

OOJG

(d)6m2

12 12

Rhombohedral Rhombohedral

G

OOJ.2..P OOJ.2..P

..m

4(d)6.. 4(d)6..

4(d)3.2

and and

(n) (n)

G

2(c)

P6x2z

12 12

c c

c

p

2(a)622 2(a)622

0^

OOcP

2(a)62m 2(a)62m

2(b)?m2

6/mmm

OOJ-P

(m)min2

(b) (b)

.

00}P6xx"

E3xx2y

..2

c c

C C

.2.P .2.P

P P

P

(k) (k)

P

12 12

2(a)Jm. 2(a)Jm.

l(a)

2(b)J.m 2(b)J.m

00^

2(b)6/m. 2(b)6/m.

6(1)

P6xx

/mcm /mcm

j j

3

P6/mmm

P6

P6/mcc PG./mmc PG./mmc 3xy2z

c

)1

12(1

OQ£..2P

E3xy2z

c

12(o)l

P3xx2y2z

.P

3xy

12

c

.m.

OOi.2

description

(n)

P3xx2z

6

6(k)m..

00^..2P

kinds)

.

3

hexagonal

of

(m)m.

in

00|P3xx2y

(P3xx)

6m2)

)

P3x2y 6(j)

kinds)

62m,

6(1 P3xx2y

2

directions,,

c

..m

.2.

of

/

P3x2z

6(i)

6(g)

(k)mm2

OOiP3xx

symmetry

(7

(h)

directions

orientations: 3(j)

P3xx

(2

E2z

4(f)3..

System

(i)3m.

D

HOP2z

symmetry

-

4(g)

r

(4

6m2

(h)

Hexagonal

JfOP2z

4(e)3..

System

(f)6..

(g)

33.

2

P2z

.

OO^E

(d)

Table

(d)6\

P2z

2(e)3.m

Hi?

Rhombohedral

E

(e)

and

HOP

2(b)

2(c)

(d)?..

OO^G

(d)

H?P

G

2(c)

(e)3.2

(c)

HOP

2(b)6"..

(c)

OO^P

(b)6~2m

(b)

c

P

Ka)

2(a)

Ka)

2(a)32.

P6m2

P6c2

P62m

P62c Table 33. Hexagonal System (7 symmetry directions, of 3 kinds) and Rhombohedral System (4 symmetry directions/ of 2 kinds) in hexagonal description

6mm - C 6v

P6mm P[z] G[z] N(z] P6x[z] P6xx[z] P6x2y[z] C 6v" 183

1 (a) 6mm 2(b)3m. 3(c)2mm 6(d)..m 6(e).m.

P6cc G c [z]

2(a)6.. 4(b)3.. 6(c)2.

P c [z] GC [Z] ,.2P c 3x[z] ..2P c 3x2y[z]

2(a)3.m 4(b)3.. 6(c)..m

Etz] .2.P E3xx[z] .2.P E3xx2y[z] c c 2 (a) 3m. 2(b)3m. 6(c).m. -177

-178

-179

-181

-182

c D

D

6

D,

D

D.^-180

D

"Q2x2yz

c

..P

1

3

(m)..2

OOJP6xx

description

P6xx

6(1)

kinds)

3

(k).2.

00-lP6x

hexagonal

of

in

E3xx

~Q2xx~

c

..2 c

P6x

6(j)

..P

1 u

kinds)

6(h)

OOr.2.P 2

directions,

00|3

of

N2z

/

3x

.2.

c

lx2yz

c

symmetry

~Q

..2P

6(g)

cc

(7

Cc

G2z

directions

4(h)3..

2p

.2P

r

i-

2

D

3

-

System

E2z

4(f)3.

(g)222

~Q2x symmetry

622

C

(4

.P

6{g)

1

3,.

N

Hexagonal

3(f)

(P2z)

System

4(e)3..

33.

~Q2z

6(f)2..

P2z E

C

2(e)6..

>

Table

002

6(b)..2

{d)3.2

P22

<

6(e)2..

Rhombohedral

OOiG

(d)3.2

and

E

2(c)

(d)222

OOJ'Q

2(c)

"Q

3(c)

P

2{b)3.2

j

C,

00

(b)622

(b)222

00^

c

C

P

Ha)

3{a)

2(a)32.

22

5

2

3

P622

P6

P6 -175

-176

,

description

E3xy2z

c

.P

2(i)l

1

kinds)

3

00j2j.

hexagonal

of

in

N2z

E3xy

6(i)2.

.P

kinds)

6(h)m..

2

2j.

directions

of

G2z

4{h)3..

symmetry

6{g)l

(7

6h

directions,

C

0(HN

(g)2/m..

3h

-

C

-

System

E2z

6/m

4(f)3..

6

N

symmetry

3(f)

2(g)

{4

.

Hexagonal

(P2z)

4(e)3..

P2z

System (f)?.

2(e)6..

33.

.

(e)

OO^E

Table

(d)?.

00}G

(d)

Rhombohedral

ifjp

2(c)

(c)

G

2

and

.

(c)

HOP

2(a)6~.

(b)

OOfP

(b)6/m.

OO^P

p

(a)

P

l(a)

/m

3

P6/m

P6

P6 -168

-169

-170

-171

-172

-173

6

C

Cc

C

description

kinds)

3

hexagonal

of

in

kinds)

6(d)l

2

P6xy[z]

directions,

of

/

symmetry

(7

directions

6

Q2xy[zJ

C

"Q2xy[z]

E3xy[z]

C 6(c)l

c c -

P

System

6

6(c)l

N[z]

symmetry

3(c)2.

3j..P

2j..P

(4

Hexagonal

System

33.

Table

Qtz]

2(b)3..

G[z]

3(b)2..

Rhombohedral 2(b)3. E[z]

and

lxy[z]

c

Q

6(a)l

[z]

P[z]

c

c c

Vi

3(a)2..

p

P

2{a)3.. P

P6

P6i

P6;

P6,,

P6o

166

165

- -

1-167 1-167

3d

3d"

D D

D

E3xx2yz

c

.cP

description

OOi.

kinds) kinds)

3 3

E3xx E3xx

P6xx

hexagonal hexagonal

in in

of of

6(h)..2

CU.2.P CU.2.P

kinds) kinds)

OO^P6x

G2z G2z

4(h)3..

2 2

3x2yz 3x2yz

directions, directions,

of of

6(g)l

36(f)l

3m)

6(g)

P6x P6x

OOJ.n.'R OOJ.n.'R

OOlN OOlN

symmetry symmetry

6(f).2.

3ml; 3ml;

(g)..2/m

(7 (7

directions, directions,

E2z

(Tim, (Tim,

4(f)3.. 4(f)3..

c c

OO^N OO^N

(f).2/m. (f).2/m.

N

N N

System System

3(f)

D._ D._

6(e)T 6(e)T

symmetry symmetry

- -

(4 (4

3m 3m

c c

c

N N

(P2z)

3(e) 3(e)

4(e)3.. 4(e)3..

OO^M OO^M

Hexagonal Hexagonal

P2z P2z

(d).2/m.

c c

System System

M

2(e)3.m

4(d)3.. 4(d)3..

33. 33.

18(d)T

' '

(.2.GElz)

(d)3.2 (d)3.2

Table Table

OOJG OOJG

(d)3.2

c c

• •

Rhombohedral Rhombohedral

- -

(c)3..

E E

'(R2z)

• •

(P2z)

12 12

2(c)

4(c)3.. 4(c)3..

and and

P2z P2z

G G

2

2(c)

c c

-»• -»•

OOi'R,, OOi'R,,

2(a)32. 2(a)32.

6(a)32.

2(a)3.2

- -

OO^P OO^P

OOJP OOJP

(b)Im. (b)Im.

(b)3*.m

.

c c

P

P P

'R

P P

6(b)T. 6(b)T.

2(b)J.. 2(b)J..

Ka)

2(b)3.. 2(b)3..

Rlc Rlc

R3m

P3ml P3ml

P3cl P3cl

P31c P31m P31m 157

-160

3

3v

3v-

3v

C/-156

C

description

6(e)l

P3xx2y[z]

kinds)

3

hexagonal

of

in

/

kinds)

3xy[z]

c

2

6(d)l

directions

P3x2y[z]

of

..2P

3(d).m.

P3xx[z]

3m)

symmetry

31m;

(7

directions,

(3ml,

..m

System

3

C

symmetry

3(c)

P3x[z]

E3xy[z]

-

c

6(d)l

(c)3..

HOPtz]

(4

(c)3m.

HOP[z]

3m

,2.P

Hexagonal

System

33.

Table

G[z]

(b)

2(b)3..

Rhombohedral

(b)

HOP[z]

E[z]

HOPfz]

and

[z]

c

(a)

P[z]

c

R[z]

Ka)

Pfz]

l{a)3.m

P 2

'R

2(a) P

3(a)3m.

6(a)3..

P3ml

P31m

P3cl

P31c

R3m

R3c

-155

-152

-153

-150

-149

°-154

3

3 3

D

D, D,

03 03

D, D,

D

be

would would

while while

, ,

(c_) (c_)

description

and and

U) U)

OO^P3xx

kinds) kinds)

3 3

hexagonal hexagonal

in in

of of

positions positions

P3xx P3xx

in in

kinds) kinds)

2 2

directions, directions,

shift shift

of of

HOP2z

no no

32)

be be

Qlxx"2yz

Qlxx2yz

321; 321;

symmetry symmetry

+

(h)

Qlx2yz

c

+

(7 (7

J|OP2z J|OP2z

directions^ directions^ c

..P

..P ..P

would would

1

2

6(c)l

(312, (312,

..P

6(c)l

2

i3

3 3

00|3

D

there there

System System

00§3

2(g) 2(g)

- -

P2z P2z

; ;

symmetry symmetry

32 32

(4 (4

3(e)

P3x P3x

Tables Tables

Hexagonal Hexagonal

(f)3.2 (f)3.2

IsJP IsJP

System System

33. 33.

Qlxx

edition.

Qlx

(e) (e)

c c

.

.P .P

.2.GE1Z .2.GE1Z

2(d)3..

HOP HOP

.P

. .

R2z R2z

1952 1952

2

6(c)3.

Table Table

International International

00g3

the the

of of

0013! 0013!

Rhombohedral Rhombohedral

(d)

and and

P2z P2z

2(c)3..

follow follow

edition edition

(c) (c)

(b)32.

OO^R OO^R

1935 1935

tables tables

(b)32.

the the

(b)

Our Our

OOJF OOJF

(a)

of of

. .

3 3

00^ 00^

(a)

R R

3 3

by by

setting setting

the the

12

21

2l

l

2

1

shifted shifted

*In *In

P3

P312

R32

P321

P3 *P3 1 o Table 33. Hexagonal System (7 symmetry directions, of 3 kinds) and Rhombohedral System (4 symmetry directions/ of 2 kinds) in hexagonal description

I - C 3i (111; II)

P311 P OOiP P2z .2.GElz N OO^N C 3t-147

Ka) 2(c)3. 2{d)3.. 3(e)

R3 R OOJR R2z M OO^M C 3i- 148 3 (a) 6(c)3. 9(e) (d)T

3 - C 3 (311; 31)

P311 P[z] JIOPlz] fiOPfz] P3xy[z] -143

Ka) (b) 3(d)l

P3, 3 1 ..Pc~Qlxy[z] C 3 -144 3(a)l

P3, -145

R3 RCz] C 3 -146 3(a)3.. 148

-1 -161

-155

-146

3

D

3v 3v

C

Si' Cq

description

rhombohedral

in

kinds)

table,

2

of

preceding

the

directions,

in

symmetry

.2

(4

(e)

co-ordinates

R.

J*

System

6(d)I

letter

J

3(e)

hexagonal

with

lattice

Rhombohedral

I2xxx

4(c)3.

the

2(c)3.

P2xxx

34.

described

with

are

Table

2(a)32

begin

(b)3m

(b)32

groups

symbols

I

P

[xxx]

space

l(a)3m 2(b)3~.

I

2(a)3. P[xxx]

Ka)

P[xxx]

their

same

The

where,

P3m

P3c

P3m

P3c

P32

P31

NJ

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-

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1

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4(e)..m

(f)222.

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4(m)2.

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2(e)

OJO(..2CIlz)

C2z

35.

4(d)2..

(c)222.

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(d)42m

HO(P2z)

I2z

4(c)2..

2(a)

2(c)2.mm

(c)

OOJP

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4(k)

.

c

(b)

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I2x

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C

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8(g)l

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c

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(f).m.

C

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- -

4xy[z] 4xy[z]

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8{d)l

8(d)l

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..m

2xx[z]

4(e)

c

8(d)l

P4x[z]

..mC4xy[z] ..mC4xy[z]

4(d) 4(d)

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Tetragonal Tetragonal

35. 35.

4(d)..m

P4xx[z] P4xx[z]

[z]

c

4(c)..m

O^O.b.C2xx

Table Table

4(c}2..

0}OC

}OOC[z] }OOC[z]

2(c)2mm. 2(c)2mm.

lzl lzl

2.mm

c

(b) (b)

^OOC[z] ^OOC[z]

(b)2.mm

3JOP

(b)4mm (b)4mm

HOPtz] HOPtz]

Ez] Ez]

I[2

c

C[z] C[z]

Ka)

2(a)

2(a)4.. 2(a)4..

P{z} P{z}

2(a)2.mm

P

8(a)2..

cm

bc

mc

2

P4bm

P4mm

14mm

P4 P4

I4cm

P4cc

P4

P4nc P4 -89

-90

-95

-92

-98 -94 -97

9

u

L

10

D

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(n)

(m)

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8(h)

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8(f)

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2.22

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422

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8(f)2..

(e)

05-0(C2z)

(f)2.22

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2(g)

P2z

OOI4

OOJ4

(o)..2

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I2z

35.

2(e

C2z

00^

4(e)4..

^O^C

4(d)2..

(f)222.

Ix

c

OJO(..2CIlz) I

c

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cc

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Dlxx2yz

2xz V

V

(b)

^OOC

2(e)

c

OHi

I .

(d)222.

.

3

8(b)l

4{d)2.22

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4(n)

4

X

3

2

4 4

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v I2z i

5

0;0..2CIlz

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c 8{c)2.

(d)422

2(c

0^0

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4{c)222.

(c)

Ix

c

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2x

c 00|I

OO.C

(b)2.22

00^1

cc

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c Dlxx

(b)

-P (b)422

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4(a)

1

(a)

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2(a) 2(a)

00j|4i 4(a)

2

1

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1

3

2

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P4.22 1422

P4

tn cn C^-84

4xyz

c

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8(k)l

OO^m.

I2xy

c

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.

J

00}P4xy

4

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3

OI2z

1

0.

of

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symmetry JJJP

4(g)

8(f)I

(P2z)

C

(5

2(g)

-

4/m

System ab

I2z

4(e)4..

(f)2/m..

P

*

4(e)2..

2(e)

OO

Tetragonal ab

OJO(..2CIlz)

OHc

2(e)

4(d)

35.

T

®^

(d)2/m..

Table

00-

C

(HOC

2(c)4..

4(c)2/m..

(c)

8(c)

OJOl

2

Ol0..2CIlz

4(c)

(c)

c

p

I

OOJl

OOJC OOil

(b)?.. ^°

(b)4/m.

(b)

(b)2/m..

00];

)

c

a

D

(a)

P

I I

C

Ka)

2(a)

2 4(

4

T

/n

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2

in

2

in

0

P4/m

P4

0

0 P4/n 14/m

P4 I4j/a Table 35. Tetragonal System (5 symmetry directions, of 3 kinds)

4" - S L P4 P 'HP P2z 0-}0..2CIlz P4xyz -81 Ka) (b) 2(e) 2(g)2.. 4(h)l

14 I 00^1 I2z I4xyz 2 (a) (b) 4(e) 8(g)l

4 - P4 P[z] 0!OC[z] P4xyfzJ Ka) 2(c)2.. 4(d)l

fP4,

4(a)l

IP4 Q

-78

HOPc [z] OJCl[z] 4 1 ..P c i2xy[z] 2 (a) (b) 4(d)l

14 0?OC c [zJ I4xy[z] -79 2(a)4.. 4(b)2.. 8(c)l

14, v Dfz]

4(a)2..

- -

2-48 2-48

l

2h

D D

P2y P2y

(o)

(D.2.

100 100

P2y

(k)

4 4

(n)

2xz

04 04

3

a.

.P

m.

(m)

2 2

P2y P2y

(j)2.

.2

2x 2x

.2.

2mm 2mm

p

(h) (h)

(I). (I).

(I) (I)

OH

c c

I2z

4(i)

4(k) 4(k)

directions) directions)

(k)

.m.

(x) (x)

(P2y)

4(g)

2x2yz 2x2yz

222

c

(j).2. (j).2.

|00l2y |00l2y

symmetry symmetry

(h) (h)

.m?

00}P2x

(3 (3

8(r)l

OOi- OOi-

I2y I2y 4(i) 4(i)

D

(f)rran2

(g)

System System

- -

.m

2xy 2xy

, ,

c

mmm mmm

(h)2.. (h)2..

OO^I2x OO^I2x

2..P 4 4

(f)

4(g)

C2x C2x

Blz

a

4(g) 4(g)

I2x I2x

Orthorhombic Orthorhombic

2xz2y

2(e)

a

(f)T

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(e)

i00.2.P

36. 36.

8(1)1

2 2

(f)l (f)l

Z*lF Z*lF

m. m.

(

(t)mm2

ab

a

Table Table

2 2

4(e)

c

JjOP

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HcF HcF

4(e) 4(e)

(d).2/m.

(e)

O^OP O^OP

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(s)

(d)..2/m (d)..2/m

(o) (o)

iQOP2z iQOP2z

Blz2y

? ?

a

222 222

0-}0(P2z)

a a

OO^C OO^C

0-^01 0-^01

2 2

(d) (d)

(d)

4(k)m. 4(k)m. 2.P

(d)222 (d)222

(c) (c)

00

(r)

(c)

c c

^00.

(c) (c)

(c)

^O^C ^O^C

00^1 00^1

(n) (n)

(q)

;-iO(P2z)

(b)

2 2

(b)

P2z P2z

2xz 2xz

(b) (b)

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(b) (b)

}00l }00l

a

c c

(j).m.

a a

c c

(a)

(a) (a)

p

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C C

P

(P2z)

2 2

2(a)

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2 2

4(m)

2 2

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Pmmm 0 0

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2x2yz

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8(k)..2

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8(h)

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8(f).2.

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a

F2x

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(f)..2/m

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(f).2/m.

8(1)

2

JJOF

4(e)

(d)I

O^P

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(b)

2

4(e)

ab

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8(c)

(k).2.

4(a)

JJO(P2z)

8(m)..2

0(HF

(b)2/m..

(b)

^OOC

(d)2/m..

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INDEX

Anorthic system, 12 Face complex, 1

Asymmorphic, 23 Form, crystal form, 1, 4 General --, 4 Axis special --, 4 alternating --, 3 co-ordinate --, 4 Hauptgitter, 18 Neben --, 3 Zwischen --, 3 Hemisymmorphic, 23

Blickrichtungen, 3 High symmetry point (HSP), 3, 17

Complex, 1 Isogonal polyhedron, 6 lattice -- (LC), 11, 85-131 Weissenberg complex, 18, 69-83 Lattice, 1 - complex, 1,11 Configuration, 11 - symmetry, 3

Co-ordinates Lattice complex (L.C.) condensed description of, 8 general --,11 special --, 11 Crystal limiting —, (L.L.C.), 11 - class, 8 invariant --, (I.L.C.), 12, 35, - form, 1, 6 40, 44, 47, 50 - symmetry, 3 variant --, 16, 38, 42, 48, 51 generating -- (G.L.C.), 16

Degradation cubic --, 9, 64 Limiting site set, 4 hexagonal --, 10, 66 Multiplicity

Distribution symmetry, 17 - of a lattice complex,"11 - of a site set, 8

Dome (=dihedron), 7 Nodes, 1

175 Parallelohedron (=pinacoid), 7 Space groups, 133-167 characteristic -- of a L.C., N-pointer, 3, 6 11, 12 N-punkter N- punktner, 3 Sphenoid (=dihedron), 7

Splitting, 5, 54 Position, 2, 11 partial --, 5 point --, 2, 11 complete --, 5 Prime Subgroup relations prime sign on Wyckoff letter, 19 - in point groups, 9, 34 - in space groups, 23, 169-173 Punktlage, 2

References, 26 Symbols Hermann-Mauguin --, 3 oriented --, 4 Representation - of a lattice complex, 11 Symmetry standard --, 11 - direction, 3 alternate --, 12 - element, 3 shifted --,14 - operation, 17 congruent —, 14 enantiomorphous --, 14 Symmorphic, 23 Rhombicuboctahedron, 8 Synonyms of G.L.C., 17 Shift vector, 14 Transformation of co-ordinates, 4 - and 3 - index symbols, 9 Site - set, 3, 6, 29, 55 Triclinic (=anorthic) system, 12 - symmetry, 3, 11, 16, 62 general site set, 4 Truncations, 8 special site set, 4 generating site set, 5 Weber four-index symbols, 9

Snub cube, 8

176 ERRATA p. 17, Sn. 6.4, line 4: in front of space group insert characteristic 9 - p. 133, Im3m(0, ), second line of entry, in 6(c).3m: replace 6 with 8 p. 138, P6^/mcm(D., ), position 4(e)3.m is correctly given. [Note erratum J b in I.T. (1965, p. 303); instead of 4(e)6 read 4(3)3m] p. 150', P4/nmm(D, ), first line of entry, above (h)..2: * replace 0|0 with OOs g p. 150, P4/ncc(D,, ), second line of entry, after 4(c): replace 4.. with 4.. 2 p. 154, P4bm(C, ), first line of entry, above 4(c)..m: replace Oj0.b.C2xx with 0|0.b.C2xx[z] p. 155, P4 ? 2 2(D, ), second line of entry: replace 4(c) with 4(c)2., replace (d)2. . with 4(d)2.. 4 p. 158, Pban(DZn0, ), ————under Pban —————————————add choice of origin: — 0 in 222 p. 165, C2/c(C? ), second line of entry: replace (b) with (b)l replace (c) with 4(c)

Remark.- Some of the conventional position letters used by Wyckoff (1930) have occasionally been interchanged in Vol. I of the I.T. (1952 and reprintings) . The original Wyckoff letters have been used in the present lattice-complex tables. Agreement with the I.T. can be achieved by means of the following changes. p. 105, under P, at P , 2, 42m, P4 2 /mcm: replace (a) with (b) replace (b) with (d) [Note that (b) should have read (c)] p. 105, under P, at P , 2, m.mm, P4 /mem: replace (b) with (a) replace (d) with (c) p. 108, under I, at F , 8, 4.., 14 /acd: replace (b) with (a) on the next line, at 2.22, 14.,/acd: replace (aj with (b) p. 150, P4 2 /mcm(D4h °) : interchange (a) and (b); likewise (c) and (d) p. 151, !4 1 /acd(D/ , ^u ): ———————interchangea — (a) ——and (b) p. 152, P42c(D2d2 ): replace (g)(h)(i)(j) with (j ) (i) (h) (g)

Wyckoff, R.W.G. (1930). The analytical expression of the results of the theory of space groups. Second edition. 'Carnegie Institution of Washington.

177 FORM NBS-H4A (1-71) U.S. DEPT. OF COMM. I. PUBLICATION OR REPORT NO. 2. Gov t Accession 3. Recipient s Accession No. BIBLIOGRAPHIC DATA No. SHEET NBS MN-134 4. TITLE AND SUBTITLE 5. Publication Date May 1973 Space Groups and Lattice Complexes 6. Performing Organization Code

7. AUTHOR(S) * Performing Organization W. Fischer, H. Burzlaff, E. Hellner and J.D.H. Donnay 9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. Project/Task/Work Unit No.

NATIONAL BUREAU OF STANDARDS DEPARTMENT OF COMMERCE 11. Contract/Grant No. WASHINGTON, D.C. 20234

12. Sponsoring Organization Name and Address 13. Type of Report & Period Covered Final Same as No. 9 . 14, Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16* ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant bibliography or literature survey, mention it here.) The lattice complex is to the space group what the site set is to the point group- an assemblage of symmetry-related equivalent points. The symbolism introduced by Carl Hermann has been revised and extended, A total of 402 lattice complexes are derived from 67 Weissenberg complexes. The Tables list site sets and lattice complexes in standard and alternate representations. They answer the following questions: What are the co-ordinates of the points in a given lattice complex? In which space groups can a given lattice complex occur? What are the lattice complexes that can occur in a given space group? The higher the symmetry of the crystal structures is, the more useful the lattice-complex approach should be on the road to the ultimate goal of their classification.

17. KEY WORDS (Alphabetical order, separated by semicolons) Crystallography ;* Crystal point groups ; crystal structure; lattice complexes; site sets; space groups.

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