Mm 1W 18 072 if "^^^"^ 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 Geologic, 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, Secreiory

NATIONAL BUREAU OF STANDARDS, Richard W. Roberts, Dnecior

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. C13,44:134). 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 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 Spl itting 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

V Occurrence of Lattice Complexes 25 Cubic Invariant 85 Univariant 88 Bi variant 92 Tri variant 93 27 Hexagonal and Rhombohedral Invariant 95 Univariant 98 Bi variant 101 Trivariant 103 28 Tetragonal Invari ant 105 Univariant 109 Bivariant 114 Trivariant 117 29 Orthorhombic Invariant 119 Univariant 121 Bivariant 124 Trivariant 127 30 Monocl inic 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 1 58 37 Monocl inic 165 38 Anorthic (Triclinic) 167

Appendix

43 Subgroup Relations 169

Index 175

Errata 177

VI . ;

SPACE GROUPS AND LATTICE COMPLEXES

Werner Fischer, Hans Burzlaff, Erwin Hellner, and J.D.H. Donnay

INTRODUCTION

A lattice is defined, sensu striata, 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-

increased in complexity, the t.{uvw) = ua + uIj + wc, tures that were worked out where a, b, c are three non-coplanar vectors and word lattice came to be used for any set of equi- u, 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 (s.s.), 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 the vectors t.{uvw) 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 crystal! bgraphically 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 the 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 crystallo- 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 lattice". 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 ^ 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 ar'e re- how unfortunate this concept of interpenetration was. peated by the lattice.

1 . . . . - ,

of crystal structure, pointing out that this analogy In 1953, in a paper given at the Erlangen meeting

is not complete. He also shows the importance of of the Crystallography Section of the German Mineral

lattice complexes^ 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

^{.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.

the 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 uni variant 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 tne symbol of the space group of higiiest rapporteurs

symmetry that contains tne lattice complex as a point The fundamental feature of a lattice complex; position'^ of highest symmetry; the space-group symbol which was pointed out by Hermann in 1953, is that its

being followed by the conventional Wyckoff letter of points either are themselves invariant or are dis.-

this 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

tr I vari ant ones to symbolizing sets of equivalent points in a point

group. Part I, accordingly, deals with site sets

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); Smimova a point group go througii the origin, symmet ry- re 1 a ted

(1966) ; Smimova 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 ^ Point position or, 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) la*, with h, k, I integers; they are the co-ordinates

It corresponds to the German PunktZage 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 Intematicmal y = v'b, z = w'a, where u' , v' , w' 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.

7 .

PART I. SITE SETS

1. GENERALITIES valent symmetry directions. Example: in the hexa- gonal system, the three columns of the symbol refer principal direction (or primary axis); 1.1 Symbolism and definitions to: (1) the The lattice, the infinite assemblage of points (2) the lateral axes of the 1st kind {Neben axes or 2nd that geometrically expresses the triple periodicity of secondary axes); and (3) the lateral axes of the The a crystal, is the basis for Bravais' definition of the kind {zwisahen axes or tertiary axes). most sym- crystal systems. It possesses, at any one of its metrical point group is the holohedry, 6/m 2/m 2/m or nodes, a point symmetry which is one of the 32 crystal ^/mmm for short. The symmetry directions of the lat- point groups. There exist seven such lattice sym- tice are vJiiat Carl Hermann called t\\e. Bliokriohtungen~ 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 synmetry or holohedry. symbols (Table 10) are explicited as 311 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, 5 and 3 are its merohedries; any substance with crystal as a whole. a rhombohedral lattice can only have, as crystal sym- A orystallographia 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

5/m 2/m 2/m; it has eleven merohedries (down to its one of the 32 crystal lographic point groups and can, position^ ogdohedry 3); any substance with a hexagonal lattice therefore, occupy a in this point group. A can have, as crystal symmetry, any one of these site set will be called an H-pointer (after the German twelve point groups, which include the five trigonal H-Punktner or H-Punkter) .. Every site of the N-pointer symmetries is invariant under certain symmetry operations, which

The Hermann-Hauguin symbolism rests on the funda- constitute a subgroup of order N of the crystal lo- 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 jgroup. axis of proper or improper rotation; in other words, a A site thus lies on one, or more than one, synmetry rotation axis^ 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

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. ^ See Introduction, last footnote.

3 Examples. A site set is provided by the corners in morphological crystal lography.^ of the cube. The eight points have the following co- A limiting site set can occur only in a special ordinates in 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 site set (see Tables 9-15).

They constitute a set of equivalent sites in 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 Bliakriahtungen that are not used. In the hexagonal

3m (of order 6) in mZm and 3 (of order 3) in both crystal system, for instance site synmetry m may

m3. In all point groups, each site thus 432 and three 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 api- 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, in Hermann-Mauguin symbol of the site synmetry, simply the first example, by each of the six complete permut- by omitting the dots. By taking the dots into account, ations of syz. These points are equivalent sites in the elements of symmetry of the site are identified point group nQm, 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 2mm. is the tetragonal way of the latter position is called the limiting site set. writing the orthornombic mmZ; in both symbols the

Example. In point group 23, the co-ordinates may 2-fold axis is directed vertically, that is, parallel be made to obey the conditions y = z = the with z. general position ixyz; then simulates the special Systems of co-ordinate axes.- The co-ordinate position :xxO: of point groups n^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 (i 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, b, a, parentheses. ^ 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 ^ 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 all In Table 9 (point group 23) the parentheses around limiting cases, while "general form" applies to 12xx 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 f9nn whose faces are oblique to every element of symmetry; any other form is a special symmetry of position :a:j/z : , where 6z2xy is realized, form. remain the same in column :xa;Q : , where (12xx) occurs.

4 . .

for the two settings 42m and 4m2; the co-ordinate axes /'i(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 4m2, 42m, 3ml, 3lm and 3m (which is treat- crystal system, the smallest cell is used and desig- ed in rhombohedral co-ordinates), the splitting point

is to a nated P (formerly called C) ; as in the tetragonal made stay in mirror, which is either normal system, provision must be made for two possible set- to, or contains, tiie principal axis. tings, 5m2 and 62m. The rhombohedral crystal system A(c). In tne remaining point groups (Table 1,

is treated twice: once in rhombohedral co-ordinates, column 1), the 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 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 2 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 (lOTl) 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. The 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 pdint 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,

spl its 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 syimetry of the final Examples: 2xy2, 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 ^ 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 set, 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 of In crystal morphology the cube, or hexahedron, is the point group. the crystal form that corresponds to the six-pointer A(a). In T, point groups 4 and 3, complete 5z . 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!

5 . .

A. If the site symmetry contains a single element 2.rm, 4.., 2..); 4xx {m.mZ, ..2); in the rhombohedral

(m, 2, 3, 4 or 6), only one more splitting, by means point groups, 2xxx (3^, 3.), 6xx (.2). Note that the of this element, is needed to bring the site synmetry hemimorphic symmetries, 2, 4, 3, 6, mmZ, 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), lz4xx, 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), lz6xx, 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, as ple: splitting 2x follows splitting Iz in mmZ) , In 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- re la ted 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 arystal form. The lines crystal lographic 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! ographic vertices are equivalent, and the polyhedron is called point groups, a number of special positions have been isogonal (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^ 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 raltiplicity. A site set of multiplicity N

4xxx (.3m, .3.), 12xx (m.m2, ..2); in the hexagonal is called an W-pointer. It is easy to see how the point groups, 2z (6mm, 3.m, 3m., 6.., 3..), 6xx (mm2, crystal -form polyhedron gives rise to the co-ordina- tion polyhedron and to coin a name or provide a des- ..2)i in the tetragonal point groups, 2z {^mm, Zrm. , cription for the latter.

See Sn. 1.1.

6 . . . .

I 2.1 The One-pointer.- The only one-pointer is the and their duals are as follows: TRUNCATED TETRAGONS,

lOINT (Fig. 1), corresponding to the MONOHEDRON (or coplanar with 0 (Fig. 21) or not (Fig. 22) [DITETRA-

pedion) GONAL PRISM and DITETRAGONAL 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. 2Z) , PRISM (Fig. 24), and TRAPEZO-

LINE SEGMENTS, one of w.iicii is col linear witli the HEDRON (Fig. 25), each with the PINACOID, correspond-

origin (Fig. 2) and corresponds to the pinacoid (or ing to TETRAGONAL SCALENOHEDRON , TETRAGONAL DIPYRAMID

paral lelehedron) , while the other is not collinear with and TETRAGONAL TRAPEZOHEDRON, respectively; t\\e RECT-

the origin (Fig. 3) and corresponds to the DIHEDRON^ ANGULAR PARALLELEPIPED (Fig. Zi,) [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

Thei-r duals are, respectively, the trigonal prism and the cubic system, have the following designations.

t(ie 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) [dihexa- 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

(dila^ , 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 PINACOID (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)^ ; CUBE WITH TWO

phenoid) (Fig. ^0) , TETRAGONAL TETRAHEDRON (ordis- TETRAHEDRA (Fig. 35) [TETRAGON-TRITETRAHEDRON)'^ ;

phenoid) (Fig. 11), and REGULAR TETRAHEDRON {Ft g. 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 (Fig. 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 si xteen-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 tne CUBE {or hexahedron); the TETRAGONAL DIPYRAMID.

TRIGONAL ANTIPRISM {V-ig. 18), whose dual is the RHOM-

BOHEDRON (in crystal -form language, a trigonal anti- 2.9 The 24-pointers.- There exist six 24-pointers.

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)^ ; RHOMBICUBOC-

TRIGONAL TRAPEZOHEDRON WITH PINACOID (Fig. 20), whose TAHEDRON (Fig. 42) [TETRAGON-TRIOCTAHEDRON)^; CUBE

duals are the TRIGONAL DIPYRAMID and TRIGONAL TRAPE- WITH TWO TETRAHEDRA (Fig. 43) [HEXATETRAHEDRON) \

ZOHEDRON, respectively. TRUNCATED OCTAHEDRON (Fig. 44) [TETRAHEXAHEDRON) ; CUBE WITH OCTAHEDRON AND DIHEUHEDRON (Fig. 45)

2.6 The Eight-pointers.- The seven eight-pointers [DIDODECAHEDRON); SNUB CUBE (Fig. ^6) [PENTAGON-TRI-

OCTAHEDRON) .

The single term dihedron replaces the two terms used by Groth, dome and sphenoid. A crystal form 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) ^ Two two polyhedra belong to the same site set

6z4xx (Table 2) .

7 . -

2.10 The 48-pointer.- The only 48-pointer is the hedra^ , 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 HEUOCTAHEDRON. 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 (<) 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 {-n) 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-j, 6.2, 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: {xys)\ = xyz; groups (so-called crystal classes)^. 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-y, 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 centrosymmetry; 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 , 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, 42m, 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- + (+++, +— k) {xyzTi), 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, 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

Chapter III in Int. Tab. (1935) J.D.H. Donnay, E. Hellner and A. Niggli (1964).

8 ±z)-n, is {001} and tne lower monohedron {001}, condensing would be ±{x, y , ±2; x, y, which 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: 4mm{B.) and 4mm(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^ 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 xj/js 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 lographic 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 oq^jz 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 the 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 [an/Jz]. 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 inmediate; 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 groups, the first three, so that xyjz becomes x~ j , y- j , in cubic orientation, in the first column and

(0.) 2. Example: 5T42 gives 5-4, 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 = 5142. 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 ' Or the symbol of the limiting site set, which is

for 2 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^ 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 tiie 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 tri variant; special

symmetry whose operation involves a translation com- lattice complexes can be bi variant, uni variant 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 lattioe 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 the 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 lattioe 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 Pmim 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 Pm2>m 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. The lattice complex of only one special position,

The positions that belong to the same lattice namely, position 48(/'):a:0^: in laZd is not a limiting

lattice complex: it cannot occur as limiting case in

^ 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

11

501-563 0 - 73 -2 the characteristic space group; nonstandard represen- which remains invariant in spite of this choice of

tations win 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 P\ as position (a), with site symmetry 1 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 P^[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 anorthia (which is used in-

position -.xyz: and is symbolized P[xyz]. The co- stead of triclinia 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 course, unless there is danger of confusion.

12 CNj iro I— IfO CM ICO r— iro O CM ICO CM 1— in I— ICM I— ICM I— ICM

I— KM I— ICM r-ICM .— ICM I— ICM I— ICM

roico LTJIOO

r-.ico r^ico

r-^ico r— i<=i- O0i

CM ICO •—t<:j- CM iro I— ICM roiCO I— iro IT) ICO CO ICO

r-ICO

CO w

I— 1^ 1— ICM r-l'::^- r— ICM r— ICM

r— 1-:^ CMiro iro

13 Distinction between a lattice complex and its re- obtained from R by a 180° rotation around the a axis

presentations.- We may distinguish between the con- (in hexagonal co-ordinates); 'S is derived from S by

and various represen- a screw rotation of power 1 or cept of the lattice complex the 3 around a 4i 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 those that were proposed by Carl Hermann (1960). tal system is preserved. The transformation matrix

1j' The new symbols were agreed upon at the Kiel Symposium that expresses the cell edges a', , c' in terms of (Donnay et al., 1966). They can be remembered without the edges a, o, 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 multipl icity w Examples: P^^ (Table 26) is a representation of P

combine to form a new complex with multiplicity 2m, with b = b'/2, so that two standard representations of

the complex letter is usually^ retained but an asterisk the complex occupy the volume of the cell. The trans-

follows it in superscript. Example: J* (pronounce: formation matrix^ 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 4m 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

pi us 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 pi us complex will ca.rry 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: illj is a shifted re- twice the volume of an I cell, which is oriented at in 13'^ presentation of J; ^^^F and p^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-

Exairples of exceptions: F + --- F is not de- felicitous to describe this transformation as a 45°

signated F* , it is called P2 instead (see below);

P + P is I, not P*. 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 •11. J + J' = J*,

While it is true that, if the o length is kept con- F + F' = P^ (note that F* is not used).

stant, this peculiar operation will transform the P + J = F, P' + J' = F', = small cell into the large one, it is also true that I + J* the same operation will not turn an I cell into an F F + F" = D, F' + F' = D',

cell! The symbols proposed by Carl Hermann make use hP + G = hPy = H, hP + N = hP. ^1^2- 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 1^. 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 The 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 AlBg

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 ^^^/^^^/OOl 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, ^^^.) The atoms of oxygen in perovskite multiplicity 4, instead of 8, and the complex symbol occupy the J complex. will be ^D. M Complex that results from the repetition of the

A mnemonic rule may be found useful: to go to a equi points of N by a rhombohedral lattice, in a larger cell, the co-ordinate axes a-] and are ro- hexagonal cell. The Ni atoms in Ni2Pt2S2 occupy tated counterclockwise, that is, to the right, and the the M complex. subscript u 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-^ and 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. complex.

In the hexagonal system (Table 29-30) one addi- S This complex has site symmetry 4 and contains tional symbol is needed whenever the o axis of the c tetragonal tetrahedra that have 4 synmetry, complex in standard orientation is to be multiplied by in Schoenflies notation, formerly called spheno-

3 to give the e' length of the cell. P^ indicates hedral . The Th atoms in ThjP^ illustrate the S this transformation. indicates the doubling of o. complex. c ^ It follows that P(,^ stands for a multiplication of c T Tetrahedra sharing all four corners. The Cu by 3x2 or 6. atoms in MgCu2 ^^'-^P'^ ^ ^ complex, and so do the oxygen atoms in Cristobal ite. Relations between invariant lattice complexes. This complex has site symmetry 222 and contains

Using the abbreviation P' to mean iUp. F" to mean rhombic tetrahedra that have symmetry 222, or V 111 333 444 ^» ' ^° ""^^^ 444 F, and similar abbreviations in Schoenflies notation. The Ca atoms in garnet F', J', D', 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

In the invariant complexes + ^Y' = """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 viae versa, so as to

form the letter "Y". The Li atoms in LiFej-Oo 6.3 Rules for selecting the generating lattice com-

occupy the Y' complex. The Au atoms in Ag2AuTe2 plex (GLC).-

occupy *Y*. The K atoms in high-temperature The problem may be stated as follows: Given a

leucite, K(AlSi20g), 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

Ca2Al2(SiO^)2. and W* accommodates the B atoms in points will by splitting produce the given variant Zn40(B02)g. 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 P5z. 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 F around each of the two points of I. number of degrees of freedom

The direction in which splitting is to take place n F=0 F=l F=2 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 24 432, 43m; 2/m 3; 6/m 2/m 2/m for site sets in point groups. Examples: P12xx is a 16 Mm 2/m 2/m univariant complex composed of points that lie on the

12 23; 622, 62w; 6/m ; 3 2/m 6mm bisectrices of the interaxial angles and thus are the 8 422, 42m; 4/m; 2/m 2/m 2/m 4mm — — apices of a cuboctahedron. ISxxx is also univariant, 6 6; 32; 3 6; 3m — — but the length x is plotted three times in succession, 4 4; 222; 2/m 4;mm2 — — parallel to the three unit vectors a , a2 and a^, so -j 3 3 — — that the resultant lies along the body diagonal of the 2 T 2 m — cell and, within the cell, the complex consists of two 1 cubes, one around each of the two points of I, the

generating complex. In the orthorhombic system (Table Example: In space group PmSf?!, position 6(e) xOO 27), F2y2z is made up of four site sets 2y2z, one . around each of the four points of the generating com- is the characteristic position for a univariant lat- tice complex. This LC may be described from two dif-| plex F; it is a bivariant complex, because y and z both of have different independent numerical values. In the ferent generating complexes, which are invari- 1(a) same system, a simple trivariant complex is P2x2y2z, ant lattice complexes: P in with site symmetry mjn 2{d) 4/mmm. the symbol of which is self-explanatory. and UL in with site symmetry The

16 . , . ) -

order of the group of site symmetry is 48 in 1(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 uni variant lattice complex will be written P6z. Example: For 8(d) in -r2i2i2^, 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 chosen rather than .2. in 4(Z)) or ..2 in 4(c). provide, for the variant LC, alternate descriptions based on different generating complexes that belong to Rule 7: If the site symmetries are one and the the same crystal system (see Tables 25-31). Two cases same point group in only one orientation, give two or can occur: the site symmetries are different point more GLC symbols, writing first the GLC obtained by groups or they are the same point group. The follow- setting x = y = z = Q in the co-ordinates of the ing rul-es indicate which of the synonyms must be se- given variant position. (If there are more than tv;o lected in order to get as short a symbol as possible. generating complexes, list all but the first one in

al phabeti c order.

Rule 2: If the site symmetries are different Example: In the example given for rule 5, there point groups of the same order, take the smallest num- were two positions with site symmetry 222., namely ber of degrees of freedom. 2(a) with the ILC and 2(c) with 0^01. The sym-

Example: To describe the tri variant LC that is bol ..2 P^I2x2yz of the complex of tne general realized in the general position 8(e) in Pbam, we position 8(p) in P4222, therefore, makes use of two choose site symmetry T in 4(a), 000, with no degree of generating complexes, of which P^ comes first. freedom, in preference to 2.. in 4(c), a-, 1/4,0, In Taoles 18-24, which relate the lattice com- which has one degree of freedom, and . .m in 4(ci) plexes to the Weissenberg complexes, symbols are given x,y,]/i, which has two. for all synonyms. Synonyms that violate rules 2-6 are

written in full and placed between brackets. If the

Rule 3: If the site symmetries are different same site set can be used after any of the generating point groups of the same order and rule 2 does not complexes, tne synonym is given in abbreviated nota- apply, take tne highest Laue symmetry (see Table B and tion. Example: S*[V*]2x instead of S*2x[V*2x]. note that 6/m arbitrarily takes precedence over 3 2/m) Remark.- The shift vector that is placed in

Example: For 16(n) in PA/mao, site symmetry 422 front of the LC symbol to designate a shifted represen- in 2(a) is chosen in preference to 4/m in 2{b) because tation of the LC will always refer to the first GLC it has a higher Laue symmetry. when there is more than one in the symbol.

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 P'i/nbm, site symmetry 422 The generation of a variant lattice complex may be in 2(a) takes precedence over 42m in 2(e). visualized as consisting of the following steps: (1)

the splitting, into a set of equivalent points, of the

Rule 5: If the site symmetries are one and the HSP of the point group that belongs to the space group same point group in different orientations, take the under consideration; (2) the subdivision of this set coordinate axes of the LC parallel to those of the into a number of subsets to be distributed in the cell space group, rather than parallel to the diagonal so that each one becomes associated with one (or more directions than one) point of the GLC. The subsets may have more Example: For 8(p) in P4222, site symmetry 222 than one orientation. oriented 222. in 2(a) and 2(c) is preferred to 222 The symmetry operation that carries a subset from oriented 2.22 in 2(e); the Neben axes are taken near the HSP to its assigned location near a point of rather than the Zwisohen axes. the GLC is represented by the symbol of a symmetry ele-

ment of the space group.' Such a symbol will stand for Rule 6: If the site symmetries are one and the the first power of the symmetry operation if the sym- same point group in different orientations and rule 5 metry element is of order 2; if the order is greater does not apply, the x direction takes precedence over than 2, the power to be used will be explicitly stated

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 I centered space group, the distribution symmetry

columns of the Hermann-Ilaugui n symbol. It will be should bring 000 onto ^^O' "ot onto OOg. 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 C^ 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 Caamim) and ibam {k) . This situation arises

group. from the fact that C may be constructed as C + Ooic 1 C /L The following rules have been applied in express- or as I + 00^1. 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 syirmetry 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 x, 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 j4^32 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-

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 . . Y*3xx and 42-. Y*3xx lead to 402 lattice complexes. The Weissenberg com- 4i for the two uni variant lattice complexes realized in plexes include all the invariant lattice complexes and

24(?) and 24 {h) , each of which retains the chirality also some univariant, bivariant and trivariant ones.

of its GLC: "Y* and "^Y*, respectively. 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 ool 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 symbol and write: plexes; if we were to give all possible synonyms, all

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

18 -- . . <

7.2 Tables of Weissenberg complexes. 2i2^..FYlxxx in P2i3. In this example, position (a) The Tables of Weissenberg complexes (Tables 18- of P2i3 has co-ordinates xxx\ [^Oj + {xxx)'\Tt. 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 a; = 0, F; for x = 1/8, \; for x = 1/4, F; for

space group, data pertaining to the position it oc- X = 3/8, 'Y; for x = 1/2, 555 F; for x = 5/8, cupies in its standard representation (multiplicity, 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 a cell edge. a constant vector. ^ In such a case, the Wyckoff There are 7 invariant complexes: P, G, E, N, 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(i'); the co-ordinates given in the In- Examples of non-shifted representations are il- ternational Tables (1952) under -^432 (i) are those of lustrated (Fig. 55) the shifted representation 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; 22^) + (xxx; ...). that these triangles do not give the complete 2-di- mensional 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 a co-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 z 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 = P^), I (alternate re-

There are five uni variant cubic lattice complexes presentation F = ly), D (which occurs only as ^D) , T that are Weissenberg complexes (Figs. 52-53). They (which occurs only as ^T). Both and ^T are used as are drawn in standard representation, each in its char- 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 a/a dicate the range between the two or three limiting ratio must be taken f 1. Additional representations cases that are used as generating complexes. Example: C, C^, ^D, and ^T do not appear on the cubic drawings and are presented in Fig. 58 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- 7.35 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 AND THEIR OCCURRENCES first generating complex is thus specified and the 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 a cell edge. followed by its alternate representations. If needed

The orthorhombic invariant lattice complexes are: a footnote lor a column gives the transformation ma-

P, I, F, D, T, which result from easily visualized de- trix^ 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: V^^, the the tetragonal system (Table 28), the multiplicity is

1 P, P , orthorhombic analog of the cubic ?2 (Fi9- 50); C^, for 2 for and C, 4 for P and C . c ab c tne orthorhombic analog of the tetragonal (Fig. 58); The next column gives the site symmetry, ori- and additional representations P^, P^, P^^^, P^^, ented in the symbolism of the relevant crystal system.

P,., 1.. I.. A, B, A, , A . B, (Fig. 61). Example: in the orthorhombic system (Table 29), the ao a b c b a b

There are 5 univariant orthorhombic Weissenberg site symmetry of P^i^ is 222 in Cmma{a) and Crma{b)

complexes, 2 bivariant arid one trivariant (Figs. 62- 2/m.. in Cmma{o) and Cmma(d) , .2/m. in Crma{e) and

63). Two of them need special explanations. Crmaif), ..2/m in Cmrm{e) and Cmmm{f) , and T in

The bivariant complex 1.2, B, A,FI Ixz, in Pnma(c) Pbanie) , Pban{f) , Pmmn{a) and Prmn{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 aa 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-^Z-^{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. 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^^ and A, I, fraction bars. Example: letting a', S' , 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). a = a, b = b, c = 2c - a will be represented by the matrix 100/010/102.

20 - -

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 that tnere are cases where the LC is defined by means and in the smaller cell, whereas we define U and T of a shifted representation. Example: in Paom, 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 yo/^^O/OOl 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 P4232, 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(;!), and .3.W2x, in 12(i). In 1432, both

For every space group, we give, for each position, I12xx, in 24(/2), and . .P^3y.x, in 24(i), 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 P3^^2, the the same multiplicity and the same oriented site sym- co-ordinates of the sites of position 3ia) were given metry, the multiplicity is written in front of the as follows, in the 1935 edition of the International first Wyckoff letter and the site symmetry, behind the Tables: second. This typographical presentation retains the X, z, 0; X, Zx, 1/3; 2x, x, 2/3. rule, followed by the International Tables, that the In the 1952 edition, the z 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 P3i21 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 PMmaa, systems: it helps bring out subgroup relations. note that P2z occurs in 4(^) as (P2z) . This should

For the sake of brevity, only the standard lat- not be written Pj,2z: the subscript must modify the tice-complex symbol is given - no synonym, no abbre- whole LC, which is P2z, defined by position l{g) in viated notation of synonyms. The shift vector, of its characteristic space group P/^/mmm. course, cannot be dispensed with. The positions are In PMmno, position 8(^)..2, the LC is written listed in the alphabetical order of the Wyckoff let- 0^^( .b.C2xx)^. Note that the distribution symmetry ters, with a few exceptions. In Pm3, for instance, it comes from the characteristic space group PMmbm. The is clear that position {h) must inmediately follow subscript c, being attached to the parenthesis, re- position (e), since the two positions accommodate the calls the fact that the LC inside the parenthesis was

same lattice complex P6z, (e) in its standard repre- defined in another space group; in other words, that

sentation and [h) in a shifted representation. the space group in which it occurs here is not its

21 cnaracteri sti c space group. position 8(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 2 value, symbolize those that would be designated by the same and that we have 2=0 in the characteristic space letter. Example: In P4/nce, 4(e)4.. is described group. It follows that the 2-fold axis (..2) must move with the complex. as oioC . .2CIlz)_. (The complex has been defined in

1 FMnrm, position 2(e)4mm, as O2O..2CIIZ.) If we set In the orthorhombic system, we keep the holohedral

2=0, 1/8 and 1/4, successively, in the co-ordinates: space group in the same setting as in the 2nd edition for hemimorphic 0^2; ^Oi; 0,^,^+2; ^,0,^-2, we obtain three possible of the International Tables (1952), but a setting retains the generating complexes: ^^1^,0^. This fact is not ex- space groups, we select that plicitly expressed in our symbolism, not even by the same systematic omissions and, therefore, shows the synonyms of the standard symbol of the LC. subgroup relations. Example: Space group 02^^^ is

In hemimorphic space group P^^ma, positions described in setting Prma. Space group Z^^^ is des-

4(e). m. and 8(/)l correspond to the LC symbols cribed in setting PmaZ in the International Tables, setting to facilitate comparisons, but we ..2P^I2x[z] and . .2P^I2x2y[z]. In both of these we keep this the distribution symmetry ..2 comes from P^„/rma, also give an alternate setting, Pm2a.

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. The 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. Al), we consider the to the description of positions by means of lattice cubic asymmorphic^ space groups and shpw subgroup re- complexes, whenever the positions have no degree of

lations connection F2-^3 to its minimal supergroups, freedom. Examples: the relation between 1^ and P^, 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 Im3m. In a second tively, is an example of Rule 1; the complex I, which example (Fig. A2), the filiation goes from Fddd, occurs as 2(a) in Im3m and as 2(a) in Pm3n, illustrate

through its minimal supergroups and the supergroups of Rule 2; and the relation between in (Pw3n)2(e) and the latter, to the common supergroup P>i2m. In pre- V* and S*, which occur in Ic3d 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 PZ-jS. A space-group symbol into complexes of lesser multiplicity. Example: enclosed between parentheses and followed by a sub- 4.. Y**2xxx in Za3d and 22.. Y*2xxx in J4i32 (Table script 2 indicates that a block of eight cells, two in 39).

each of the three co-ordinate directions, is being lb. The site symmetry of the invariant position

does not remain unchanged; then both the splitting

L space group is called symmorphia if it contains number and the invariant complex cnange. Example:

a position whose site symmetry is the same as the 4.. Y**2xxx in Ja3d and 4.. l2Y**lxxx in J?3J (Table

point group isomorphic with the space group; hemisym- 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

asymmorphia, if neither of the above. Examples: Pm3m change in the naming of the generating complex; then

is symmorphic, Pw3n is hemisymmorphic , Pm3n is asym- the variant lattice complex remains the same. Example

morphic. Site symmetry m2m is available in Pm3m, I5z in Im3m, ]2{e)4m.m, remains I5z in Pm3n, 12(/)ot7)2.

432 in Pn3n, but neither m3w nor 432 is site symmetry (Table 40).

in PrnJn. 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 is 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 Ia3 and Fm3. In

pletely different generating complex. Example: Table 42 two space groups, f4^32 and J4^32, have a

4a.. Y**3xx in Ia3d{g) and 22.. P26xyz in Ia3(e) common subgroup P^^^^i the solid lines show where the

(Table 40). special positions of f4^32 and I4i32 are to be found Relationships are shown between asymmorphic cubic in the subgroup. Finally, Table 43 shows the relation- body-centered space groups, which have the same cell: ships between positions in different crystal systems,

Ia3d, Il3d, 14^32, Id3, i"2^3 (Table 39). The series illustrated by the space groups Fd3m, Fd3, I^-^/amd

Jm3m, Pm3n, Ia3d, Ia3 , 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;

'^3m 43m 432 m3 23

i-(Jw3m)

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

Donnay (1960) Tables of magnetic space groups. Mineralogische Gesellschaft, Sektion fiir 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 ., 113, 142-154.

II. Special positions. Jour. Phys. Soc. Japan Internationale Tabellen zur Bestimmung von Kristall-

J7(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.il nommene kri stall ographische 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 . 130 , Ledgemont Labora- A24, 57-67. tory, Lexington, Mass. Burzlaff, H., W. Fischer and E. Hellner (1969) Bemer- Loeb, A.L. (1957b) 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 . 1,

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., .A25, groups in Miller axes. Acta Zryst 715-716. 1924, pp. 168 et ss.; 3rd ed. , 1941, IPP. 275 et ss. Donnay, J.D.H., E. Hellner and A. Niggli (1964) Niggli. P. (1928) Kri stall 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. 12S, 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 viergl iedrige Zonensymbol des

complexes. Acta Cryst . 3g, 703-712. hexagonalen Systems. Z. Krist . 57, 200-203.

Hermann, C. (1935) Gitterkomplexe. Int. Tab, zur

Bestimmung von Kristal Istrukturen , vol. l,

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. 5. 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 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. C\±ie. 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 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- tri tetrahedron. 35. Cuboctahedron. Rhomb-dodecahedron. 37. Pentagon-tritetrahedron & 2 tetrahedra. Pent agon- tri tetrahedron 38. Dihexahedron & octahedron. Dihexahedron. 39. Truncated tetragonal prism & pinacoid. Di tetragonal dipyramid. 40. Truncated hexagonal prism & pinacoid. Dihexagonal dipyramid. 41. Truncated cube. Trigon-trioctahedron. 42. Rhombi cuboctahedron. Tetragon-trioctahedron. 43. Cube & 2 tetrahedra. Hexatetrahedron. 44. Truncated octahedron. Tetrahexahedron. 45. Cube & octahedron & dihexahedron. Didodecahedron. 46. Snub cube. Pentagon- trioctahedron. 47. Truncated ciiboctahedron. Hexaoctahedron.

o

32 33 .

Order of

Fig. 48. The point group a- ; the lower end of a connecting

line is a subgroup of that < it 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 I — K U —' —

- — H\ -04-

- r- 4- -c)- 4 - () 04

0- -0 i\- \- 04- 4 I

0- 4- •0 —1 —1 —1

1 1 - 3 -h-

0 1- 1- _ e

- - - 1 -3 0- 4 •0 1 _l 1 D

1 1— — 1 — 1 1 1 4^26-n4 1 1 - - -3- -5--7- -1 - -5--7-1-3- IT 1 0 -t- -f- H--H -H —h- -h - -H - -3- - 1 -7- -5- -7- - 3- - 1 -5- 26 — 0- 0 — 26 -h— H- -h -\- H- -4- - - -5- -7- - 1 -3- -3- -7- - 5- - 1 0- -+- -+- — -\- —\— —\—

-7- -5- -3- - 1 - -5- - 1 - -3- -7-

1 1 1 h^^^26 — 1 1 1 1 1 T

37- 15 •37 TV 6^ ho- 4-| [-4 — 0-1-4 oH ho 4-^0—4-1 6 15 -37- -1 5-

0-H4 ho 4-hO— 4-| |-4 0^4 2 2 6 2 6 37 15' L •15 37 V

The 16 cubic invariant lattice complexes in standard orientation. Heights in eighths of cell edge.

35 0- 0 04- 04' 04

o o b c. b

04- 04 04

co - cr> oc

0- -0 04- 04 04 " J- ±± p 2 2 2 4 4 4^2

0- -0 04 04- 04 0- -0 •O-

26 26

04 04 04 0 0 •0

26- 26-

0- -0 04 04 04 0- •0 0- J.J_J_ 2 2 2'p

1 1 1 1 1 1 1 -15- -37- -15- -37- -37- -15-1 -37- -15- H- -37- -15- -37- -15- -15- -37- -15- -37- -h -h -15- -37- -15- -37- -37- -15- -37- -1 5- H- + -f- -37- -15- -37- -15- -15- -37- -15- -37-

1 1 1 1 1 1 1 1

3 3. 13. 3 r 1 p ±±1 p IT 4 4 4 4 4 4 r 8 8 8 "2 8 8 8 '2

— c - f 0 26 26- 26t04t26^04^26

26-04-26 04-26 04 04 04

- -C)- -4 0- -0 26 26- 26- 04 26 04- -26

26- 04-^26-^04-26 04- 04 04

1 0- •26 26- 26-^04-^26-^04-^26 111 I J J- a. a. a.

•4- 02 26- 02 0- -0 0- 04- 26 46 '46 26

-6-4- 02 --04-- 02 04-'02 46 46 46

-0 -0- 26-]-2|-|-26-^02_[.26 0 0 04 04 46

02 4-04-LQ2 .02 46 46 04- 46

0- -0 02 0 04 26^°i^26^b/f46 46 ^26 i.± j_r) D 2 2 2"^ J"

Fig. 50. Cubic invariant lattice complexes and representations, Heights in eighths of cell edge.

36 — r — T

-26 - 1 I 1 1 — 1 r — 1 -1- -3^-5--7- -5- -7- -1 —-3- -\- -\-^ -h- -4— -3- -1- -7--5- -7- -5- -3- -1- -0 — 26 26 -H —1— -+- -5- -7- -1- -3- -1 - -3- -5- -7-

1 -h -H -h —1— 1 -7- -5- -3- -1- -3- -1- -7- -5-

1 1 1 •26 1 1 1 1 1

W T 2 2 2'

7- r 5 I "TI I 2 -6 4q ho -0- 6

6 0-h4 h4- oH -0- 4- 0 2 _L v V

15- •37- 37- •15 13 .1 3 4-r26 57 5 -26 -^--|-26 -0- |_0 — 4 + 0 — 4-| [-4 OH |-04 04 -[-04 --04 -j H 6- 26 26 -26 -0 -15 15 ,1 3 .1 3 26 26 1-37 + 37-H I-+57 57 H 26 26 26 i - |_4 —0^4 0-| ho 4-|-0 |-04--04-|-04— 04-| 2 6 26 H—I—|- 26-1—I—1-26 4-1- 26- -15- •37- 37- 15^-1 I I 13 I I I 1 3_1_J 57 57' w Wc

-1 3- -7-

5- -7

3 -7- -5-

2 2 2 ' Y 2 2 2 ^ 04T26T04-r26-r04 -1 3- 5+7- -I — 5 —7 •3- I : I T I I -+ 26404--26--04--26, ]

-7- 1 1 5- 04--26--04--26 04 7 5 -3- 3 7 — 5 -1- 26--04--26 04—26 -3 -5- -7 -5 1 3 -7 _i_ _l_ 04-L 26 -L 04-L26 -L 04 Y

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

37 •©(§)-

® D©

I2j3(^7) 2^2^.. F| Y Ixxx 12^3 (Z>) 2^3.SVlz ® UP OP, 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 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 E N

M

40 0,6 0,4,8 0,4,8

41 P6|22/2';3,2.Pc,EcQ;iXX

P<6jaJ 3,2,..Pc.,Ec^Qclxy PZm\fd) .2.GE1Z OG E Cc

P32i2/'^7; 32..PcQlxx P322I r^7; s^. . p/q Ix

OOfP, 00|+Q

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

42 PZz{a) 32..R,"^Qlxy

A 3 3^n:

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

43

501-563 O - 73 - 4 44 [7][5H (D [3][5}- ®

©

©®[SI- (3)®-

P4/nmm ic) ..2CI1Z I4/acd ie) 4..I,P^,lx

1— OO^OC DO^^I J 4a. 4a. 8a "cair

(D©- (p- ®©

mm-

ID [2]

<2>-

0) ©©

I42d(^) 4..nf^lx P4322(d7) 4,.. P T Ix 3 CC C O VT n ILL F 2 4 xc

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

45 <3>

..22^TC,,lx

'-'4 4 8 ^CC

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

46 0

0 0 0 04- 04

04 04

04- 04- •04 04- 04 0^

04 04 0

04- 04- 04 04- -04 0- 0— Pbc ac

0- 0 04- -04

0 26

04- •04

-0 0 0 0— 4 0 — 4 —

-4- 0 0—4 — 0 — 4 — 0 A

0- 0 0- 0-

0 4

0 0- •0- Aa Bb

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 CO)® 2 ©(pd?

6®C0)

-©©[2

Pmma (e) .2. Pa Biz Pmmn /"^y Zj.. CIlz O^OOPa DiO^B OC 00^1

©@ Klh {6^< C6X2

©(2)

®© >{I]- 6 he C6X2

ZxwzxT\(c) 2j..CcFly Cmma /^p^y 2.. PabFlz OOOjC O^jF

H]®©

©@-{2} Imma (e) .2. B^AJz PbcmiW 2.T Pbc Ab CeF Uy © O^OBb

Fig. 62. Five \ini variant and one bivariant orthorhombic Weissenberg complexes. Heights in eighths of o.

48 —

u •H Q) 0) s 0 "

0 • A-t 4-> M 0 0 +J m •rH

H> 0) M c P •rH

>i CO rH -P

0 CP rH ^ 0) -P

• CO rO 0)

4J 0) C fC -H s& 0 u • (0 > •H A !h • ^! (3) o Q) CD < M 0 0) 0 O g CO •H GO W +J (D •H u

49 / / / / J 4 0 4 0 4 I I I I 0 4

Fig. 54. The two monoclinic invariant Weissenberg complexes, in standard and alternate representations. Heights in eighths of h.

50 X < P2/C fej cF^Aly

X < C^/cfeJ T CcFly Oooic^ OO^iF

pz^/rr^reJ 2^ f^ACIlxz oo^op^ doHa Oiioc A^^^I

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

51

TABLES

i .,. . .

Table 1. Rules of Splitting

Point group: homopolar single, 1st Site 2nd Site 3rd Site Point-complex primary polar Split. Symm. Split. Symm. Split. Symm. symbol axis axis

1 Ixyz 1 Ixyz 1 2xyz 1 Immediate 2xyz 4 4xyz 1 4xyz 3 6xyz 1 6xyz *

m Ixz m 2y 1 lxz2y 2/m 2xz m 2y 1 2xz2y Mm 4xy m. 2z 1 4xy2z ** Z/m 3xy rn. 2z 1 3xy2z 6/m 6xy m. 2z 1 Stay in m. 6xy2z <4 m 2 4xz .m. 2y 1 then out. 4xz2y '4 2 m 4xxz . .m 2y 1 4xxz2y (3 m 1 6xxz .m. 2y 1 6xxz2y

3 1 m 6xz . .m 2y 1 6xz2y i 3 m 6xxz .m 2y 1 6xxz2y

2 2 2xz 1 ly2xz 4 Iz 4.. 4xy 1 lz4xy 0 V Y/ J- i IZ J. jxy 1 izoxy 1 Ixxx 3. 3yz 1 lxxx3yz 6 Iz 6.. 6xy 1 lz6xy 2 2 2 2x 2.. 2yz 1 Stay on axis, 2x2yz 4 2 2 4x .2. 2yz 1 4x2yz (3 2 1 3x .2. 2yz 1 then off. 3x2yz

3 1 2 3xx . .2 2yz 1 3xx2yz, *** (3 2 3xx .2 2yz 1 3xx2yz 6 2 2 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 2z 1 Ix2y2z m 2 m ly m2m 2z m. 2x 1 Iy2z2x m m 2 Iz mm2 2x ,m. 2y 1 Iz2x2y 4 m m Iz 4mm 4x .m. 2y 1 Iz4x2y m 1 Iz 3m. 3 XX .m. 2y 1 jlz3xx2y

3 1 m Iz 3.m 3x . .m 2y 1 Iz3x2y 3 m Ixxx 3m 3y .m 2z 1 Ixxx3y2z

6 m m Iz 6mm 6x . .m 2y 1 Iz6x2y

m m m 2x 2mm 2y . .m 2z 1 2x2y2z 4/m m m 4x m2m. 2y m. 2z 1 4x2y2z

1 3/m 2 m 3x m2m 2y m. 2z 1 3x2y2z t 3/m m 2 3 XX mm2 2y m. 2z 1 3xx2y2z 6/m m m 6x m2m 2y m. 2z 1 6x2y2z

mm3 6z 2mm. . 2x m. 2y 1 6z2x2y

4 3m 6z 2..Tmi 2xx . .m 2y 1 6z2xx2y m 3 m 6z 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. **

' 3/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^ 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); 43m(c) ; 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).

Di hexahedron + octahedron. Pi hexahedron (or pentagon-dodecahedron )

6z2x: ±(x,0,±z) k m3(c)

Truncated tetrahedron. Tri gon-tri tetrahedron . (For

Cube + 2 tetrahedra. Tetragon-tri tetrahedron . (For

6z2xx: (xxz; xxz)k ; xxz it 43m(d)

Pentagon-Tri tetrahedron + 2 tetrahedra. Pentagon-Tri tetrahedron .

6z2xy: (xyz; xyz; xyi; xyz)K 23(d).

Truncated octahedron. Tetrahexahedron

6z4x: ±(x,0,±z) ir m3m(d)

Truncated cube. Tri gon-tri octahedron (For Jz|<|x . .)

Rhombi cuboctahedron. Tetragon-tri octahedron . (For x|<|z| .)

6z4xx: ±(x,x,ifcz k; xxz it) m3m(e).

Snub cube. Pentagon-trioctahedron .

6z4xy: (xyz; xyz; xyz; xyz; yxz; yxz; yxz; yxz)K 432(d),

Cube + octahedron + di hexahedron. Didodecahedron .

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^ Found in the Hexagonal and Rhombohedral Systems (in hexagonal co-ordinates)

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

Point. Monohedron ,

Iz: 00 z 6mm(a) 5mm(b) 6(a); 6(b); 3ml (a) 3ml Cb) 31m(a) 31m(b) 311(a); 3lUb),

Line segment (col linear with origin) Paral 1 el ehedron .

2z: 0,0, ±z 6/mmm(a) 6m2(a) 62m(a); 522(a); 6/mCa) 6(a); 3ml (a); 31m(a); 321(a) 312(a); 511 (a).

Trigon coplanar with origin. Trigonal prism .

3 XX xxO 6m2Cb); 6m2(c). 3x: xOO' 62mCb); 321(c); 62mCc); 321(b).

Trigon. Trigonal pyramid .

Iz3xx: xxz 3ml (c). lz3x xOz' 31m(c). lz3xy: xyzj 311 (c).

Truncated trigon coplanar with origin. Pi trigonal prism .

3xx2y: (xyO, yxO) 5m2(e). 3x2y: (xyO^)^ ' S2mCe)

Truncated trigon. Di trigonal pyramid .

1 z3xx2y: (xyz, yxz) 3ml (d). ^ Iz3x2y: (xyO J , 31m(d).

Hexagon coplanar with origin. Hexagonal prism .

6x1 (±x,0.0) 6/mmm(b) 622Cb); 3ml (b) ^ 6xx: ±(xxO ) 6/mmmCc) 622 Cc); 31m(b), 6xy: ±(xyO') 6/mCb).

Hexagon. Hexagonal pyramid .

Iz6xx: (xxZt-)^ 6mmCc) lz6x: (±x,a,z}^ 6mm (d) lz6xy: (xyz, xyz)^ 6(c).

Trigonal prism + pinacoid. Trigonal dipyramid .

3xx2z: (x,x,±z) 6m2(d). 3x2z: Cx.O,±z)^ 62m (d). 3xy2z Cx,y,±z)^ 5Cc).

^ Limiting cases not included.

56 .

Table 3 (concluded)

Trigonal trapezohedron + pinacoid. Trigonal trapezohedron .

3x2yz: (xyz; ^xz)^ 321(d). 3xx2yz: (xyz; yxz)^ 312(d).

Trigonal anti prism. Rhotnbohedron .

5xxz: ±(xxz) 5ml (c). 6xz: ±(xOz)^ 31m(c). 6xyz: ±(xyz)J 511(b).

Truncated hexagon coplanar with origin. Dihexagonal prism.

6x2y: ±(xyO^) ^ 6/mmm(f).

Truncated hexagon. Dihexagonal pyramid .

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

Pinacoid + rhombohedron . Hexagonal scalenohedron .

6xxz2y: ±(xyz; yxz) 3ml(d). 6xz2y: ±(xyz^) SlmCd). ^

Truncated trigonal prism + pinacoid. Pi trigonal di pyramid .

3xx2y2z: (x,y,±z; y,x,±z) Sm2(f). ^ 3x2y2z: (x,z,±z^)^ 62m(f )

Hexagonal prism + pinacoid. Hexagonal di pyramid .

6x2z: (±x,0,±z) 6/mmmCd). 6xx2z: (xjXjiz^-)^ 6/mmmCe). 5xy2z: +Cx,y,±z)| 6/mCc).

Hexagonal trapezohedron + pinacoid. Hexagonal trapezohedron .

6x2yz: ( xyz, xyz, yxz, yxz )^ 622(d).

Truncated hexagonal prism + pinacoid. Dihexagonal di pyramid .

6x2y2z: ±(x,y,±z^)^ 6/mmm(g).

57 . .

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

Co-ordination polyhedron. Qua! crystal form . Si te-set symbol . Co-ordinates. Occurrences.

Point, iionohedron ( pedion )

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

Line segment (col linear vn'th origin). Paral lelehedron ( pinacoid )

2xxx: ±(xxx) 3m(a); 32(a); 31(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) 5m(b).

Truncated trigon. Pi trigonal pyramid .

Ixxx3y2z: xyz tt 3m(d).

Trigonal trapezohedron + pinacoid. Trigonal trapezohedron .

3xx2yz: (xyz, xzy)K 32(d).

Trigonal anti prism. Rhombohedron .

6xxz: ±(xxz k) 3m(c). 6xyz: ±(xyz <) 31(b).

Pinacoid + rhombohedron. Hexagonal scalenohedron .

5xxz2y: ±(xyz v) 3m(d).

Limiting cases not included.

58 . . .

Table 5. Site Sets^ in the Tetragonal System

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

Point. Monohedron ( pedion )

Iz: OOz 4mm(a); 4mm(b); 4(a); 4(b).

Line segment (collinear with origin). Parallelehedron ( pinacoid )

2z: ±(00z) 4/mmm(a); 42m(a); 4m2(a); 422(a); 4/m(a); 4(a).

Tetragon (square) coplanar with origin. Tetragonal prism .

4x: +(xOO)t 4/mmm(b); 42m(b); 422(b). 4xx: ±(x,±x,0) 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; xxz^ _ _ 42m(c). 4xyz: xyz, xyz, yxz, yxz 4(b).

Truncated tetragon coplanar with origin. Pi tetragonal prism .

4x2y: ±(xyO; yxO)^ 4/mmm(f).

Tetragonal tetrahedron + pinacoid. Tetragonal scalenohedron .

4xz2y: ±x,±y,z; ±y,±x,z^ _ 4m2(d). 4xxz2y: (xyz; xyz; yxz; yxi)^ 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/mmm(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. Pi tetragonal di pyramid .

4x2y2z: (x,y, z; y,x,±z)^ 4/mmm(g).

^ Limiting cases not included.

59

501-563 O - 73 - 5 , ) ;., , .

Table 6. Site Sets^ 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, z mm2 ( c lz2x: ±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 di pyramid .

2x2y2z: ±(x, y, ±z; x, y, ±z) mmm(g)

Limiting cases not included.

60 .

Table 7. Site Sets^ 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 (pinacoid) .

2y: ±(OyO) 2/m(a). 2xz: +(xOz) 2/m(b).

Line segment. Dihedron .

Iy2xz: xyz, x^z 2(c). Ixz2y: xyz, xyz m(b).

Rectangle coplanar with origin. Rhombic prism .

2xz2y: ±(x, ±y, z) 2/m(c).

Table 8. Site Sets^ Pound in the Anorthic (Triclinic) System

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

Point. Monohedron ( pedion ).

Ixyz: xyz 1(a).

Line segment (collinear with origin). Paral 1 elehedron ( pinacoid )

2xyz: ±(xyz) T(a).

^ Limiting cases not included.

61 o CN X X X X X X g X CN X g X g X X • • X X X X X • <:t- g

o • o N • IM • g X X • N rg X X X to CM l£) CM X X rv) X CN ^ g

O o N N N N N N N N CM CM CN CM CM CM X X X X X a" CM X e X g X • X • X X n X n X X <^ X N N N X 00 • • CO • CO • N CM X CM CM N X CM X N X X X X X g X X <^ g X X X N o X X N E >> (/) N +-) N N X X X X CM X CM CM N X > X g X oo • CM ^ • g r- . ^ e >> u) o o CM CM 1 CM CM CM , (U X X .H X X H >> • X 4J H X • (/> i- X Q.-f- g =! (/) X X X O X X X X X 0) S-T3 X g X g CM h- M >> >> ^ • CM • N M .H CM >, CM CM M +> N • >^ CM N X >1 CM M lO CM rH N X CM X CM X ^ >> rH >> r c (U X X N X X X •1— •r-' O S- Q. O CM C Q) to •I- CM CM ^+J X X X "=ME c >> X • X CM O -Q ^ • N .-I CM N •1— N g N +J -o WD •1- nj Q. to c a. >. 1- e >> o o >> CM o o g X X g X X • X o CM X >> CM X CN X CM to to CN t/i o X lO CO ^ lO X X X X (JD g CO g g to • .-1 -Q CM . (U N N CMM CM I/) O N O) o I X I 4J I— X IX 1 X 1 X r-) • •- o X IX X 1 X Cv] X • X • X cn • Ut -Q X g lo e X to • ro • to 00 g lO • -— E ' g E >> I/) CMI E S CM 00 CO ro CO ^3 CMIE CM g o g' o N N N • • I N N N CM C\J c^ n (U J CM ID CM to CM to CM cn r—

N N IM IM CM IX N N I X CM CM CM X CM 1 X CM 1 X X 1 X 1 X to o X g X 1 X g to E X rH X rH X rH N rH X to X N to to OO to CO

N N N IM X N g M CM X g CM CM CM to If) CM • CM O X to X rH X rH X rH N rH >> H X • X to N to to OO i.ri to ro

>> >> o >> • >> • CM • CM CM cn CM • • • CM 1 X X X rH CO ^g' X g X g X g to to to rH X to C>0 00 to OO cu N M M CM CM CM >> CM N IM >> >> CM X >» CM OO CM . . ' CM I X to CM H > H >,rH to rH X X X IM X X X N to oo OO to

OMI I E CM CM CMI CM E CM 'Ol i HO ito to toi S to to

62 CM CM CM g g CM •

"x^ X CM g X CM X • CVI ,CN J CM CM

>> CM >l X CM X CM g X . X • CM X CM

N N N X N CM CM CU O CM • CM • CM • X H X H X e N g X g CM CM 10 O CM CM • >> •• CM CO >» N N CO N >> CM N CM CM CM N H CM >> . N g >. CM O N g X N g* >> CM H X X CM CM g g o CM M o S- N >. N X o CM CM CM CM N >> >5 N >> X >> N >. CM CM CM rH X N N H CM rH I CM CM CM N H N H CM rH X N X >> CM X H X X >» X CM CM CM

CMI j s CM J3 -Q CMI CM CM CMI to 1 CM CMI E E CM It—

X g X rH X g X rH X rH X H X «3 CO • X to • to CO U3 CO .

IX IX IX I X IX • IX r X CM X rH rH CNI X X 1 X X rH X H X • CO g X to CO CO u u

in x: P Eh I X I g g X I X 01 N E I X X IX X X N N g N N N N CM CM n Cvl r-" N n X to X CM to CO n CM n CM n • CM n • n to CO Q) u> tn 4J (1) (U N N ix" H -P 4J N IX IX IX CM N X 01 (S tS u N CM X X CM IX N c: a o IX • IX CO N X H CO rH X rH rH rH CO rH -H -H X g X X N rH CO IM N N g N CO CO to r— r-i Kt X - N X X X X >H M >J B X X X rH X rH X Tf O 0 X X g to to N +J to (U I I N X X! O O CM l/l o o o IX X N CM X >> to X CM IX N CO X H rH oo 9 N g N CO g X H X rH X rH N H X N CO CO to O (fl (d Eh to x; ro n c! tJ S- M o C X X X X X 0) CP ">> -a X •H >> >> X X X 0) j3 td X • • CM CM '>J CM g ^ g X X X x: O X C CM n 1 ro CO CO IX CM IX CM — CM CM I— ro -p 0) CM I X >> X r X rH X rH rH o X rH . . > U3 X X X JQ U3 CO CO CO CO to CO rH d •H e N rH x: a Cn o CM N U -H >> >. >> N >. > CM CM CO CO >> N IM H a M CM >> IX CM N >> X CM N X .Q H c O ro X rH . rH N Otl X X >> CM X X X rH (3 o X! IX rH IM N X H X X H CO rH . X EH CO CM I IX rH CO rH to X -H X X IM N CO to to X X X N rH to CO CO to a X) Q, H -H -H lOiroS CU U U CMI E E CM J3 O O CMI E CO E CM CMI m ICO ICO CO CO S CM "3 CO CO ICO CO CO (N CO a -a m n

63 1 , t ,

^3 N O O N N N N N N N o o CN CM CN CN CM CN

o o N N N N o o CD CD CD CD CD o o X X X o X :t o

H X X X X X X X X X X X H X X X X X X X X X X X H X X X X X X X X X X X CO J- 00 00 zt CM

,

o o 1 X 1 X 1 a 1 X X H X CD CO o X X o X X X X X X X X CM Csl X X X X ' ^ f— -J- ^ H

^3

1 H H , , X X , N X tsi X X N CM N X N X J- CM X X X J- CO ^3 X N N X X X N N X H CD CD J" X X X H X X X CD ^

o

ISl X o CM tsl J- x X X X X X N O o CM J" ^— H N N N CD CD

O 6 >, CM &H Xb X X J-

>-i CM N >, Cvl >i Sl >) >, >-> N tS] >l CN X CM CM >, CM CM N N CM CO >1 CO N X X X X >i CM N X >, CM to N X CM N >, J- CN CM CM CM tsl X >1 X X X X N N N tM X X X X ^ x" X X X X CD CD CD CD CD J- J- CD CO CD

6

o

E e CM f-, 100 oo CO ICO CO

E CM g CM +-> CM 1 \zt J- E IJ"

e CM 6 6 CM e CM E E CN o CO CO 00 CO CO CM e (M 100 CO 00 100 00

e Id- d- e CM J- J- 1 e 1^

64 N tsl N SI tsl N CM CM rH CN CM

o

o CN] CM CM

X CM X X CM CN

IX I X IX IX X X X X CM CN

X X X X CM X X CM N CN

I X CN X CN

tsl X N CM X CN N N X CN X X X X N X X X Csl CN

N tsl >> tsl CN CN CN tsl o ISl Csl r-{ CM

tsl X ISl N N O Csl CM CN X X X X tsl X CN CN <-) o >, CN X >1 CN X CM CN CM X X X >, X X X CM CM tH X

IM >> IM CM tsl ISl Csl CN ISl >i >. N tsl CN CN CN CN X >l N >, N X CM ISl >> X CN N ISl >, >, CM X X >1 X CN CM ISl CN CN CN ISl CN CM Csl tsl X >i CM Csl tsl tSl Kl CN X CN CN CM CN X CN X tsi ISl X tsl ISl X CN CM Csl X X X X Nl X X X X X >, X X X X tSI X X x* ^ CN

cm| e l-H

e Csl E E CM CN| E CN B CN E CN cn| e CM B 1 e CM CN cn| E l-H

cm| E e 1^ E CN e 1 CN CM CN cm| E E i CN CM .CM e CN I CM E CN e' CN CN cm| E CM

E CN CN E 1 E CN CM cm| E • CM CM E* CM CM CM i'e • e' I'H

65 o o o o X X X X X H X to CO CD CO

o o •r" o IX IX a. a. IX X X IX •r- 3 X ID ID X s- o CO o s_ I/) D1 •O 3

^3 N C ea O) X •<- (U s- ^ o tsl N IX N CN CM X tsl

IX IX UD I X CO X X N X tsl o CO •f- s- (O X s- >, CM O) •- CM 1 X 3: (u X X X X ID CO CO CD CO

-o I— c

O) N >, tsl CSI 1— (/I SI >i >> CN N t^ J2 CM CM >. CM N N >, CN 1 X CM tsl >, ^3 tsl >, >, CN X >1 CSI CM X CM X X >, CN tsl tsl CO 1— lO CM CM IX CD CM CD 1 X CO CO CM IX X X X N X xy xy tsl X X tsi N X X xy tsl CD CO CO rH CD CD CO

CM| e cm| E B CN CM cMj^e e CM CM 100 ICO CO ICO CD I e llD ICD CD ID E ICD CD I

66 o o o X o X X CM CN X CM

o I X I X o X X I X I X X CN CN X X CN

SI N o N o Nl o CN o CN

N I X N CN X CM N

I X I X CN I X IX X X ISl X X CM CN

N X N N o CM CN O N IS] X X N X X X X CN CN rH

o CN o CM CN I X X X X X X CN

N N CN >> N CM CN CN >i ISl CN >, N CN >1 >i >, >i X CN CN IS) CN CN N 0-J X CN ISl CM CN CM IX IX ISl 1 X N CN CN 1 X >, X N X X X X X X N X X X CN .H CN CN H <-! CN

CS1| E cn| e 6 S CN CN cn| E e CN e CN cn| e CM 6 e CN

67 n { ) ) K

Weissenberg Complexes

Table 18: Cubic system

P Pm3m l(a)mZm 000

P4-XXX t

-i- //VI -t-/v* /~i ) P12xx ± ( Xj ±iCj UJ K 22. .P 2xxx laZ(c) I± (xxx);{l0ll± (xxx)}k 4. .P^3xx 1432 d') { Jt (xxO) ; J+ (xxO; xOx; Oxx) } k

. r 2 " XX J-fflOffl [ U J

11 7T Jr DZ^XX ir^ Of ( L J ( XXZ y XXZ J XXZ

Pw 'K f n ) r DZZX Z ( C/j xS / K

XVm 'Km \ r DZHX Z ( ZS / TT

r DZM-XX tTnoTii ( 777/ — ( *Cj ins Kj XXZ TT /

OP C rrr O V zr^bZ/X nmoO I 7^/

P6z2xy P23(j) (xyz;xyz;xyz;xyz) k

P6z2xx2y P43m(j) (xyz;xyz;xyz;xyz) tt

PSzIxy r a::^2; xyz; xyz; yxz; yxz; yxz; yxz) k

P6z2x2y Pm3(l) ± 2/ J ±2; Xj y^±z)K P6z'4x2y PmZmin) ±(x,yj ±z;x,y,±z)T^

22. . P 6xyz Ia3(e) {J± (xyz) ; \0\I± (xyz;yzx;zxy ) }k nP 6z2xy F43o (h) {F-\-(xyz;xyz;xyz;xyz) ;\\ \F-\- (yxz; yxz; yxz; yxz j } k

m. .P 6z4xy Fm3c (j') {F+(xyz; xyz; xyz; xyz; yxz; yxz; yxz; yxz) ;

{ . .2P 6z2x2y} 5 2 \F-\- (xyz; 3[yz;xyz; xyz; yxz; yxz; yxz; yxz )}<

1 Im3m 2(a)m3m 000;

I4xxx I43m(a) I+(xxx;xxxk) I6z Im3m(e) I±(00z)k 1 8 XXX Im3m(f) I±(xxx;xxxk) 112 XX Im3m(h) I± (x, ±X, 0) K

. .2I4XXX Pn3m( e) XXXy XXX }^ J 2 2 2^ ^XXXy XXx^

I6z2xx I43m(g) !+{(xxz; xxz ) k; xxz tt } I6z2x Im3 (g) J± (x^O^±z)k I6z4x Im3m(j) I±(x, 0, ±z)-n

IBz^xx Im3m(k) I± (Xj Xj±z k; xxz tt )

. .2I6z2x Pm3n(k) ±{x,0,±z;ir2+(±z^0,x)}K

. . 2I6z2xx Pn3m(k) {xxs^ xxz y xxz y xxZy '2 2 2*^ (xxz y xxZy xxZy xxz ^ }"

I6z2xy 123(f) I+(xyz;3:yz;xyz;xyz) k I6z2xx2y Il3m(h) 1+ (xyz;xyz;xyz;o!yz)-n

I6zHxy I432(j) 1+ ( xyz; xyz; xyz; xyz; yxz; yxz; yxz; yxz) k I6z2x2y Im3(h) I±(x,y3±z;x,yy±z)K I6z4x2y Im3m(l) I± (x^y, ±z;Xjyj ±z)t\ . . cI6z2xy P43n(i) {xyz; xyz; xyz; xyz; Hi+ (yxz;yxz;yxz;yxz) }k . . 2I5z2xy P4 32(m) {xyz;xyz;xyz;xyz; \\\+(yxz;yxz;yxz;yxz) }k n. . I5z2>Qr Pni(h) {xyz;xyz;xyz;xyz; lll+(ocyz;xyz;xyz;xyz) }k n. . I6z4xy Pn3n(i) {xyz; xyz; xyz; xyz; yxz; yxz; yxz; yxz; \\\+(xyz; xyz; xyz; xyz; yxz; yxz; yxz; yxz) } k

69 ) { K ;

..2I6z2x2y PmZn(l) tfajji/j ±3;a;j^, ±z; H2+ TyjiCj ±2;yja:j ±2^}k ..2I6z2xx2y PnZm(l) {xyzjx^Sjxyz;xyss^ii+(xysjxyz;c!ysixyz)}'n

d.21^6z2-xy Fd3o(h) F-^-{xyz;xyz;5yz;5yz; 111+ (yxz;yxz;yxz;yxz) ; 5 H+ Cyxz; yxz; yxz; yxz) ; 111+ (xyz; xyz; a^z; xyz) } k

Pm3m Z(c)4/mm.m \0\k

.3.J2X Tm3(f) ±(\+x,0^\)k .3.J4X Pm3m(h) ±(l+x^0^i)T\

F FmSm 4(a)m3m 000;\0\k

F4xxx F43m(e) F+(xxx;o^xk) F6z Fm2m(e) F±(00z)k F8xxx FmZm(f) Ft (o!X)!:x;xxxk) F12XX Fm3m(h) F± (x^±x^O)k

be. . F2xxx Pa3(a) ±(xxx) ; 101±(xxx)}k

.FSxx P4^32(k') {xxO; 2(95+ rxxO; xOx; Oxx) } k

F6xx Pn3m(i' ± {xxO; r xxO; xOx; Oxx) } k

42, . F^Sxx Fd3o(g') F+ {xxO; lOl+ ( xxO; xOx; Oxx) ; H 2+ TxxO) Ti\l+(xxO; xOx; Oxx) } <

F5z2xx F43m(h) F+{(xxz; XXz ) k; xxz tt } F6z2x Fm3(h) F+ (x^O, ±z)k F6z4x Fm3m(j) Ft (x^O^tz)TT

F6z4xx Fm3m(k) Fi(XjXj+s k; xxz tt )

F6z2xy F23(h) F+ (xyz; xyz; xyz; xyz) F6z2xx2y F43m(i) F+(xyz;xyz;xyz;xyz)T\

F6z4xy F432 (j) F+ (xyz; xyz; xyz; xyz; yxz; yxz;yxz; yxz)K F6z2x2y Fm3(i) Fi(Xjyj+s;Xjyj+z)K F5z4x2y Fm3m(l) F±(Xjyj+z;Xjyj+z)-n

be. . F6xyz Pa3(d) t{xyz;\0\+(xyz;zxy;yzx)}K

111 . 25 3. P4232 4 (a). 32 8 8 8.9 8 8 8 I'

Y2xxx P4 32(c) 8 8 8 — (XXX) ; { 8 8 st (XXX } K

'Y3xx P4p2(d) { 5 H+ CxxO) ;l\l+(xxO; xOx; Oxx) } k

42.. Y3xx2yz P4^32(e) { 5 5 5 + ro^jz; yxz );lll+ (xyz; zxy; yzx; yxz; zyx; xzy) } k

Pm3n 6(c)lm. 2 t(OU)<

.3.W2Z Pm3n(g) t(0^l^l+z)K {.3. J*2z} .3.W4XX Pm3n(j)

J* Im3m 6(h)4/rm.m I+(\0\)k

.3.J*2x Im3(e) I±(1+x^0,1)k .3.J*4x Im3m(g) It(\+x^O^\)-n

70 D Fd3m 8 (a) 43m F+(000;HV

D6z FdZm(f) F+ (0, 0, ±z; 1^ l±z) k ..2DM-XXX FdZm(e) F-^{xxx;xxxk: )}

..2D6z2xx FdZm(g) F+{xxz; + (xxz: )}k

..2D6z2xy F4^22(h) F+{xyz;xyz;xyz;ayz; Hl+(yxz;yxz;yxz;yxz)}K d..D6z2xy Fdi(g) F+{3^jz;xyz;5yz;syz; l%l-\r(xyz;xyz;xyz;xyz)}K ..2D6z2xx2y FdZmd) F+{xyz;xyz;3yz;xyz; lll+(xyz;xyz;xyz;xyz)]T^

"^Y* m^32 8(a). 32 I+(lll;iU^)

22..Y*2xxx 14 32(e) I+[Ul±(xxx); {IU±(xxx)}k] 42.. Y*3xx I4^32(h) I+{lU+(xxO); iU+(xxO;xOx;Oxx)}K

22 . . Y*3xx2yz 14 ^32(i) I+{lll-\-(xyz;yxz) ; 111+ (xyz; zxy; yzx; yxz; zyx; xzy ) }k

W* ImSm 12(d)4m.2 I±(OH)<

S Ii+3d 12(a)4.. I+(OU;OU)k

.3.S2Z I43d(d) I+(0,1,1+z;0,1,\±z)k

. 3dS4xyz I43d(e) I+{Oli+ (xyz;xyz;yxz;yxz) ; OII+ (yxz;yxz;xyz;xyz) }k

"^V 14^32 12(0)2.22 I+(OH;OU)^

.3.V2Z I4^32(f) I+(0,l,l±z;0,i,l±z)K

T Fd3m 16(o).3m F+(U\;UU)

22..T3XX F4 32(g) F+{lU^(xxO) ;IU+(xxO;xOx;Oxx)}k

4..T6XX FdZm(h) F+-{lll±(xxO) ; UI±(xxO;xOx;Oxx)]k

Y** Ia3d 16(b). 32 ItdlUiU^)

4..Y**2xxx Ia3d(e) I±[lH±(xxx) ; {Ui + (xxx) }k]

{4. . I 22xxx} 4a..Y**3xx laZd(g) I+{UU(xxO) ;IU+(xxO;xOx;Oxx)}k

4a. .Y**3xx2yz Ia3d(h) I±{\\1+ (o^jz;yxz) ; l^l-\r(xyz;zxy;yzx;yxz;zyx;xzy)}K {4.2l26xyz}

S* laSd 24(d)4.. I±(OU;OU)k

.3.S*2z Ia3d(f) I±(0,l,l±z;0,l,l±z)K {.3.V*2z}

71 . )

V* Ia3d 24(c) 2. 22 I±(OU;Oli)>i

2^2^..FYlxxx P2^3 4 (a). 3. XXX; {\0\-i-(XXX ) } K

2^2^. .FYlxxx3yz P2^3(b) {xyz; \0\+(xyz;yzx;zxy ) }k

2^2^. .P^Y^lxxx 12^3 8(a). 3. J+ [xxx; { (xxx) } k)

2^2^. .P^Y^lxxxSyz I2^3(c) I+{xyz; \0\+ ( xyz;yzx; zxy) } K

2^3,SVlz 12^3 12(b)2. 5j 8+2j 5 J s~z) K

"f" (cCCCCC I43d 16(c). 3. X+ { 2C3CCCy 1+ u 4 ) ^

{ lOl+(xxx) ; lH+(xxx) }k]

. 3.J2S*V*lx Ia3 24(d) 2.. I±(l+x^O, I; l+x, ^^V <

Table 19 : Hexagonal System

(and rhombohedral system in hexagonal description)

P P5/mmm l(a)6/mmm 000

P2z P6/rrmn(e) 0^0, ±z P3xx P6m2(j) xxO;Xj2XjO; 2x,Xj,0 P3x P62m(f) xOO;OxO;xxO P6x P6/mmm(j) ±(xOO;OxO;xxO) P6xx PB/mmmd) ±(xxOjXj2x,0; 2XjXjO)

' . . 2 P 3 X P6 /mom (g ) xOO; 0x0; xxO; 00\+(xOO; 0x0; xxO

. 2 . P^E3xx P6 /mmc (h ' )xxO; x^ 2x, 0; 2x, x, 0; 001+ (xxO; 2x, 0; 2x^ x, 0)

3 . .P^+Q2x P6 22(g) ±(xOO) ;00\±(xxO) ;00\±(OxO) 32..P^+Q2xx P6^22(i') ±(xxO) ;00\±(2x,x,0) ;00\±(x,2x,0)

P3xy P6(3) xyO;y,x-y,0;y-x^x^O P3xx2y Pdm2(l) xyO;yjX-y3 0;y-XjXjO;yxO;y-Xjy,0;x,x-y,0 P3xx2z P6m2(n) P3x2z PB2m(i) x,0,±z;0,x,±z;x,x,±z P3x2y P62m(j) (xy0;y,x-y,0;y-x^x,0) t P6xy P6/m(j) ±(xyO;y,x-y,0;y-x,x,0) P5xz P31m(k) ±(xOz;Oxz;xxz) P6xxz P3ml(i) ±(xxz;Xj2XjZ; 2XjXjZ) P6x2z P6/mmi(n) ±(XjOj±z;0^x^±z;XjX^±z) P6xx2z P6/mmm(o) ±(x,x,±z;x,2x,±z;2x,Xj±z)

72 ) ) ) )

P6x2y P6/mrm(p) ±(xyOjiijX-y^O;y-x,x,0)T

z/j . .2P 3xy P6c2(k') xyO; x-y, Oj y-x, x, 0; 001+ (yxO; y-Xj y, 0; x, x-y, 0) .2.P'^E3xy P62c(h') xyO; y^ x-y, 0; y-x, x, 0; 00\+ (yxO; x-y, y, 0; x, y-x, 0) 2 ..?^E3yy P6 /m(h') xyO; y^ x-y, 0; y-Xj x, 0; 00\+ (xyO; y, y-x, 0; x-y,x, 0) 3^..P^+Q2xy ± (xyO) ;00i+ (y-x, x, 0) ; 00i± (y, x-y, 0) 72.P 6xy Pefmccd) + (xyO;y, x-y, 0; y-x, x,0);00\± (yxO; x-y, y, 0; x, y-x, 0)

. .2P'^3x2y P6 /mcm(j') {xyO; y, x-y, 0; y-x, x, 0; 00i+ (xyO; y-x, y, 0; x, x-y, 0)}t

. .2P^3x2z P6ymam(k') X, 0, ±z;0,x, +z;x,x, ±2;00l+(x, 0, ±z;0, x, ±z;x,x, +z) {m. .P6xz} .2.P^E3xx2y P6yrmo(j') xyO; y,jc-y, 0; y-x, x, 0; yxO; y-x, y, 0; x,jc-y,J); 001+ (xyO; y, y-x, 0; x-y, x, 0; yxO; x-y, y, 0; x, y-x, 0) .2.P E3xx2z P6 /mmo(k') X, X, ±z; X, 2x, ±z; 2x, x, ±z; 00\+ (x, x, ±z; x, 2x, ±z; c {m. . P6xxz} 2x, X, ±z

P3xy2z P6(l) x,y,±z;y,x-y,±z;y-x,x,jLz P3xx2yz PS12(l) xyz; y, x-y, z; y-x, x, z; yxzj y-x, y, zj x, x-y, z P3x2yz P321 (g) xyz; y, x-y, z; y-x, x, zj yxz; x-y, y, z; x, y-x, z P6xyz PZjg) + (xyz;y,xry,z;y-x,x,zj P3xx2y2z P6m2(o) X, y, ±z; y, xry, +z; y-x, x, ±z; y, x, ±z; y-x, y, ±z; x,3^y,±z^ P3x2y2z P62m(l) (X, y,_±z; y, s^y, ±z;y-x,_x, ±z ) x P6x2yz P622(n) xyzj y, x-y, zj y-x, x, zj xyzj y, y-x, z; x-y, x, z; yxz; x-y, y,j.; 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, x, ±z ) P6xz2y PZlmd) ±(xyz;y,x-y,z;y-x,x,z)T P6xxz2y PZml (j) ± ( xyz; y, x-y, z; y-x, x, z; yxz; x-y, y, z; x, y-x, z) P6x2y2z P6_/mnm(r) ±(x,y,±z;y,x-y,±z;y-x,x,±z)T

. .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 P62a(i') X, y, ±z; y, x-y, ±z; y-x, x, ±z; 00\+ (y, x, ±z; x-y, y, +z; {m..P 3x2yz} x,y-x,±z)

32..P^'^Q2x2yz P6^22 (k) xyz; xyz;x-y,y, z_;y-x,y, z_; 001+ (y-x, x^, z; x-y,x,z_;_ X, y-x, z; X, 3:^y, z) ;00\+(y, x-y, z;y, y-x, z; yxz, yxz

. .2P 3x2yz P6,22 (i) xyz; y, x-y, z; y-x, x, z; yxz; x-y^y, z; x, y-x, z; {?2.P E3xx2yz} 00\+ (xyz; y, y-x, z; x-y, x, z; yxz; y-x, y, z; x, x-y, z) 2^. .P E3xy2z Pe^md') ^, y> ±3; y, xry, ±z; y-x, x, ±z; 00\+ (x, y, ±z; y, y-x, ±z;

{m. . P^6xyz} x-y,x, ±z)

. .cP E3xx2yz P21a(i') xyz; y, x-y, z; y-x, x, z; yxz; y-x, y, z; x, x-y, z; {?.2P 6xyz} 00\ + (yxz; x^y, y, z;x, y-x,z;xyz;y, y-x, z; x-y, x, z .n.P 3x2yz PZcl(g') xyz; y, x-y, z; y-x, x, z; yxz; x-y_,y, z; x, y-x, z; {?2.P 6xyz} 00\+(yxz; y-x, y, z;x, x-y, z; xyz, y, y-x, z; x-y, x,j m. . P 5x2yz P6/maa (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 2; X, y-x, yxz; y-x, z; x, x-y, z; 001+ (xyz; c J ZJ y, 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 PB^mcmd') (X, y, ±z; y, x-y, ±z; y-x, x, +z) x; 00\+ (x, y, ±z; y, y-x, ±z; {m. .Pj,6xz2y} x-y,x,±z)T .2.P^E3x^2y2z PB^rmod') ^1 y> ±2; y, x-y, +z; y-x, x,jLz; y, x, ±zj y-x, y, ±z; x, x-y, ±z; {m. . P^6xxz2y} 001+ (y^ X, ±z; x-y, y, ±z; x, y-x, ±z; x, y, ±z; y, y-x, ±z; x-yjX, ±z)

P6/mmm 2(o)6m2 ±(\\0)

G2z P6/iwrm(h)

73 ) } ; }

P6 /mine 2(o)6m2 - ( 3 3 U/ o E2z P6^/mma(f)

R3Tn 3 fa J 3/77. 000;±(m)

R2z R3m(c) R+(00z) R3xx RZm(h') R+ (xxO; X, 2Xi 0; 2x, x^O) R3x R32 (d) R+ (xOO; 0x0; xxO) R6x RZm(f) R± (xOO; 0x0; xxO) .n. 'R 3x R3o (e') R+{xOO; 0x0; xxO; 00\\ (xOO; 0x0; xxO) c

RSxy R3(b') R+ (xyO; x-y^ 0; y-x, x, 0) RBxxz RZm(h) R± (xxz; X, 2x, z; 2x, x,z) R3xx2y R3m(a') R+ (xyO; y, x-y, 0; y-x, x, 0; yxO; y-x, y, 0; x, x-y, 0) .n. 'R^3xy R3o(b') R+ { xyO; y, x-y, 0; y-x, x, 0; 001+ (yxO; y-x,y, 0; x,x-y, 0)

R3x2yz R32(f) R+ (xyz;y, x-y, z; y-x, x, z; yxz; x-y, y, z; X, y-x, z R6xyz Rl(f) R± (xyz; y, x-y, z; y-x, x,z) R6xxz2y R3m(i) R+ (xyz;y, x^y, z; y-x, x, z; yxz; x-y, y, z; X, y-x, z ,n. 'R 3x2yz R3o(f') R+{xyzjJi, x-y, z; y-x, x, z; yxz;^x-y, y, z; x, y-x, z; .5. { 'R 6xyz} 001+ (yxz; y-x, y, z; x, x-y, z; xyz; y, y-x, z; x-y,x,z)

N P5/mmm 3(f)mrm \00;0\0;\\0

N2z P6/rmm(i) \,0,±z;0, \,±z;\,\,±z

3(c) 222 \00;0\l;\\\

Q2z P6^22 (f) 2,0, ±z; 0, 2, \-z; \, 2 J \-z

R3m 9(e).2/m. R+(\00;010;\\0)

,2.GElz P3ml 2(d)3.m ±(\lz)

P3212 3(a')..2

.P^'''Qlxx2yz P3^12(a') xyz; yxz; 00\+(y, x-y, z; x, x-y, z) 00\+(y-x, X, z; y-x, y,z)

32-. P^ Qlx P3221 3(a').2. xOO; 0x%;xx\

32-.?^ Qlx2yz P3^21(c') xyz; x-y, y;_z; 001+ (y, x-y, z; yxz ) 00\+ (y-x, X, z; x, y-x, z)

74 K

"""Q 3,.2P^ Ix P6,22 6(a) .2. xOO; xxl; Oxi; ocOj; xxl; Oxi 1 Cc c 1

"^Q 3 .2P lx2yz P6 22(a) xyz; x-y, y, z; 001+ (x-y , x, s; x, x-y^z); 00\+

{3^2.g^^E^+Qlxx2yz} (y>x-y, z;yxz) ; 001+ (xyz;y-Xjy, z) ;00l+(y-x, x,

Xi y-x^z) ; 00\+(y,y-x, z;yxz)

3,2.P- e/q Ixx P6.22 6 Cb ) » • 2 ooooO^ 2cCj 2cc^ cccc 1 Cc C c 1 6 J 3 j 2^ 23(j^ cc^ 3 J cc ^ 2cc^ 6

3^ . . Pj^"'"Qlxy P32 3(a')l xyO;y,x-y,\;y-x,x^\

''1 6(a')l xyO; x-y^ X, I; y, x-y^ \; xyl; — 2 5 2/— Xj 3 z/j y^x^ 6

Table 2o: Rhombohedral complexes in rhombohedral description

p P3m 1 (a) 3m 000

P2xxx P3m(a) ± ( xxx) P3y P3m(b') 00ZK P3xx P32 (d) xxOk P6xx P3m(f) ± (XXO) .nI3xx P3o(e ') {xxO; 1^^+(xxO)}k

P3yz P31(b') OyzK

P6xxz P3m(h) ± (xxz ) K P3y2z P3m(c') OyzT\ .nI3yz P3c(b') {Oyz;\\\+(Ozy)]K

P3xx2yz P32 (f) (xyz; xzy) k P6xyz P31 (f) ±(xyz)K P6xxz2y PZm(i) ±(xyz)-n .nI3xx2yz P3c(f') {xyz;xzy; \l\+(xzy;xyz) }k {.2I6xyz}

J P3in 3(e). 2/m '^0\k

75

501-563 0-73-6 ) ) ) ) )

Table 21: Tetragonal System

P4/mmm 1 (a)4/mmm 000

P2z P4/rnmm(g) ±(00z) PHxx P4/mmm(j) ± (Xj ±Xj 0) Plx P4/mmm(l) ± (xOO) T .b.C2xx P4/mbm(g') ±(3:x0);\\0±(xx0)

. .2P I2x P4^/mmo(Q) + (x00;0x\) .2.P^2xx P4 /mom(i) ±(xxO;xx\) .b.C*^2xx I47mom(h' I± (xxO;xx\)

P4XXZ P42m(n) xxz;xxz;xxzT P4xz P4m2(j) tXj Oj z; 0, ±Xj z P4xy P4/m(j) ±(xyO;yxO)

P4x2y P4/rnmm(p) ± (xyO; yxO) t

P4xx2z P4/mmm(r>) ± (Xf Xj ±z; Xj ±2 ) P4x2z P4/rrmn(s) ±(x^0, +z)t .P I2xy P4Mj) ±(xyO;yx\) .2 ,P%xy P47moo(m) ±(xyO;yxO;xyl;_yxl)

. . 2c5xxz P4/nbm(m' xxz; xxz; xxz t; i\0+(xxz; xxz; xxz x )

. .mC4xy P4/rribm(i) ± (xyO; yxO); \ \0± (xyO; yxO) { .b.C2xx2y} .b.C2xx2z P4/mbm(k') ±(XjXj±z);\ \0± (x^ X, ±z

. . mCUxz P4/nim('C) iXj Oj z; ixCj z; ii0+(0j ±Xj z; ±Xj 0,z) {. .2CIlz4x} m. . P 4-xz P4 ^/rmc(o' iXj z; 0^ +Xj z; 001+ ( ±Xj 0^ z; 0, tXj z) {?.2P I2x2z} ..2P I2x§y P4^/rmc(q) ±{xyO;xyO;yx\;yx\) .2.P^2xx2y P4 ^/mam(n) ±(xyO;xyl)T m. .P i+xxz P4^/mam(o ') xxz; xxz; xxzt; 001+ (xxz; xxz; xxzx) {?2.P 2xx2z} .b2C 2xy^ P4jmbo(h) ±(xyO) ;\\0±(xyO) ;OOl+(yxO) ;\\\±(yxO)

, .mC 4-xy I4/mam(k) I±(xy 0; yxO; xy\;yx\) {?2.C 2xx2y}

.b.C 4xxz I4/mcm(l' ) 1+ {xxz; xxz; xxzt; \\0+ (xxzt; xxz; xxz) } {?b.C 2xx2z} c

P4xyz P4(h) xyz; xyz; yxz; yxz P4xxz2y P42m(o) (xyz; xyz; yxz; yxz x PHxz2y P4m2('l) 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) ±(Xjy^ ±z;y,x,±z) P4x2y2z P4/mmm (u) tfXjj/j ±z;y^x,±z)T .2.Pc4x7z P42a(n) xyz; xyz; yxz; yxz; 00\+ (xyz; xyz; yxz; yxz ,P I2x2yz} c ' . .mC^xyz P42fi(f) xyz; xyz; yxz; yxz; 5 lO+ (yxz; yxz; xyz; xyz ) {. 2i.CIlz2xx2y} ..2Pcifxyz Plo2(3) xyz; xyz; yxz; yxz; 00\+ (yxz; yxz; xyz; xyz ) {4^. .P^2xx2yz}

. .2C'4xyz Plb2(i) xyz; xyz; yxz; yxz; \\0+(yxz; yxz; xyz; xyz {.b.C2xx2yz} .2 .C2xx2yz P42^2(g) xyz; xyz; yxz; yxz; 2 10+ (xyz; xyz; yxz;yxz) {..2CIlz4xy}

76 . )) ) ) ; } )

..2P^I2x2yz P4^22(p) xyz; xyz; xyz; xyz; 00\+ (yxz; yxz;yxz; yxz {.2.P^2xx2yz} m. .P^i+xyz P4jm(k') xyz; xyz; yxz; yxz; 001+ (xyz; xyz; yxz; yxz .? 2xy2z}

lCi4xyz P4/n(g) xyz; xyz;yxz; yxz; \ lO+ ( xyz; xyz; yxz; yxz) {ICIlz^xy} m, .P^4x2yz P4/moo(n' xyz; xyz; xyz; xyz; yxz; yxz; yxz; yxz;

{.2.P 4xy2z} 001+ (xyz; xyz; xyz; xyz; yxz; yxz; yxz ; yxz ..mC4x2yz P4/nbm(n) xyz; xyz; xyz; xyz; yxz; yxz; yxz; yxz;

{ .2C4xxz2y} 5 iO+ (yxz; yxz; yxz; yxz; xyz; xyz; xyz; xyz) ..mC4xy2z P4/mbm(l) ±(x,y, +z;y^x, +z); \\0±(y,x, ±z;x,y, ±z)

{ .b.C2xx2y2z} ..TnC4xz2y P4/nrm(k) X, +y_, z; X, ±y, z; y, ±x, z; y, +x, z; l\0+(±y,x, z; {..2CIlz4x2y} ±y,x,^zj±x,y^,zj±x,y,z)

. .c2C 4xyz P4/noc(g) xyz; o:yz;yxz; yxz; \ \0+ (xyz; xyz; yxz; yxz ) {?2^cC 2xx2yz} 00\+(yxz;yxz;xyz;xyz) ; m+ (yxz; yxz; xyz; xyz) {1.2(c5lz) i+xy}

m. .P^M-xz2y P4^/mmc(r' X, ±y, z; X, ±y, z; y^, +x^ z; y, ±x, z; 00\+ ( x, ±y , z; {. .2P I2x2y2z} x,±y,zjy,±x_,z;y,±x,z) m. .P 4xxz2y P4^/mcm(p' (xyz;xyz;yxz;yxz)T;00\+(xxjz;'xyz; yxz;yxz)x {?2.P 2xx2y2z}

.22C i4xyz P4^/nho(k) xyz; xyz; yxz; yxz; \ \0+ ( yxz; yxz; xyz; xyz ) {?b2C 2x2yz} 00\ + (xyz; xyz; yxz; yxz ) ;\\\+(yxz; yxz; xyz; xyz)

{ .2cC^2xx2yz}

m.2C 4-xyz P4^/mbc(i ') oryz; xyz; yxz; yxz; \ \0+(yxz; yxz; xyz; xyz)

{?b2C 2xy2z} 001+ (xyz; xyz; yxz; yxz ) ; ^l^+ (yxz; yxz; xyz; xyz) {nib.C^2xx2yz}

. .m2C Hxyz P4^/nam(j) xyz; xyz; yxz; yxz; 2 5O+ (yxz; yxz; xyz; xyz); {?2 mC 2xx2yz} 00\+ ( yxz; yxz; xyz; xyz) ; H 2+ (xyz; xyz; yxz; yxz {4. rF2xxz2y} {.2^2(CIlz)^2xx2y}

. . 2C Hxyz I4c2(i) I+{xyz;xyz;yxz;yxz; \\0+ (xyz; xyz; yxz; yxz) 2xx2yz} .b.C 4-x2yz I4/mcm(m' I+{xyz; xyz; xyz; xyz; yxz; yxz; yxz; yxz;

{?b.C 4xxz2y} 001+ (xyz; xyz; xyz; xyz; yxz; yxz; yxz;yxz) ) {.b.c'^4xy2z} {.b.c"^2xx2y2z} c '

m/mmm 2(a)4/mrim 000;\\\

I2z I4/mmm (e) I±(00z) I4xx I4/mrm(h) I± (x, ±x^ 0) mx I4/mmm(i) It (xOO) T

. n. I2xx P4 /rmm(f) ±(xxO); l\\±(xxO) .2.F 2xx I4yacd(f') I+{+ (xxO); \ \0+ (xxO); \0l± (xxO); 0\ l± (xxO) } c

IHxz I4m2(i) 1+ ( ±x, 0, z; 0, ±x, z )

IHxxz I42m(i) 1+ (xxz; xxz; xxz x ) mxy I4/m(h) I± (xyO;yxO) I4x2y I4/rnrm(l) I+(xyO;yxO)T

I4xx2z I4/rrmn ( m) It (cCj Xj tz; X, Xj tz ) I4x2z I4/mrm(n) It (x, 0,tz)i: '± . .2I4xy P4/mnc(h) (xyO; yxO) ;\\\t (yxO;xijO)

. . 2I4XXZ P4ynnm (m) 32CCZ ^ CCCCS J CCCCS T ^ 222"^ (cCCCZ j OCXS j CCCCS T ) .n. I2xx2y P4 ^rmmd) t(xyO) T;V2\±(xyO)T

77 ) ± ) ) ; }

.n.I2xx2z P4ymnm(j ) ±(x,X3 ±z); lll±(x,x, ±z)

. . cli+xz P4 yrmc (g) +x, (9j z; ±Xj z;\\\+(Oj tx^ z; ±Xj 0,z)

5. . F2xxz P4ynom(i' ± ( xxz );ilO±(XXz );lOl+(xxz );0l\+ (xxz {.2 2(CIlz) 2xx} ^ .bdF 2xy I4^od(b') 1+ { ± (xyO) ;l\0± (xyO) ;lOl± (yxO) ;0\l± (yxO) } c ^

IM-xyz 14(g) 1+ (xyz; xyz; yxz; yxz) I'4xz2y -ft I4m2 (j) ( X, +V J z ; X, ±y ,z;y,±x,z;y 3±x,z) I4xxz2y I42m(j) J+ (xyz; xyz; yxz; yxz ) x Ii+x2yz 1422 (k) J+ (xyz; xyz; xyz; xyz; yxz; yxz; yxz; yxz) I4xy2z I4/m(i) I± (x,y, ±z;y,x, ±z)

I4x2y2z I4/mmm(o) I± ( y, ±z; y,x,±z)T

. . cI4-xyz P4_2 o(e) xyz; xyz; yxz; yxz; l\\+(yxz; yxz; xyz; xyz) ..2I4xyz P4n2(i) xyz; xyz; yxz; yxz; \\\+ (yxz; yxz; xyz; xyz

{ .n. I2xx2yz} ,2^. I2xx2yz P4 2 2(g) xyz; xyz; yxz; yxz; 555+ (xyz;xyz;yxz;yxz)

n . . I4xyz P4jn(g) xyz; xyz; yxz; yxz; \\\+(xyz; xyz; yxz; yxz

. . cI4x2yz P4/nnc(k) xyz; xyz; xyz; xyz; yxz; yxz; yxz; yxz; 5H+ (yxz; yxz; yxz; yxz; xyz; xyz; xyz; xyz

. ,2mxy2z P4/rma(i) ± (x, y^J-z; y,jc, ±z)j\\l±(y, x, ±z; x, y, ±z)

. ,2I4xxz2y P4 /nnm(n) ( xyz; xyz; yxz; yxz) i;\\\\(yxz; yxz; xyz; xyz ) x •n. I2xx2y2z P4yrmm(k) ±(x,y,±zJ\;\\\±(x,y,±zjT

, . cI4xz2y P4ynmo (h) x^ ±y, z; X, +y, z; y^ +Xj, z; y, +x, z; 111+ (y, z; y,±x,^zjx,±y^,zjx,±y,z)

. 22F 4xyz I4yaad(g) xyz; xyz; yxz; yxz; \ (yxz; yxz; xyz; xyz )

{?b2.F^2xx2yz} \0l+ (xyz; xyz; yxz; yxz) ; Oi 5+ (' yxz; yxz; xyz; xyz)

D F4^/ddm 8 (a) 42m F+(000;UI)

14^/amd 4(a)4m2 I+(000;OU) V D2z I4yamd(e) I+(0, 0, ±z;0^l^l±z) ^D4xx I4^/amd(g) I+{±(x,±x^O);OU± (x, ±x, 0) }

. . d"^D2x 14 md(h') I+{+(xOO);0\l±(OxO)} .2,^D2xx 14 y 2(d) 1+ { ± (xxO) ;Oll± (xxO) }

1 V 4^. . D2x7 14 (b') I+{±(xyO);0\l±(yxO)] 1 V .2. DHxz 14 /amd(h) 1+ { ±x, 0, z; 0, ±x, z; 0\ 1+ ( ±sc, 0, z; 0, ^x^z)}

. .d^D2x2y I4^r"d(G') 1+ { (x, ±y, 0);Oll±(±y, x, 0) }

.2.'^D4xyz I42d(e) 1+ {xyz; xyz; yxz; yxz; 0\ 1+ (xyz; xyz; yxz; yxz) .2."^D2xx2yz 14 22(g) I+{xyz; xyz; yxz; yxz; 0H+ Txyz;xyz;yxz; yxz) }

a. . D4xyz I4\/a(f) I+{xyz; xyz; yxz; yxz; 011+ (xyz; xyz; yxz; yxz ) } .2.^D4xz2y I4yamd(i) I+{x, ±y, z; X, ±y, z; y^ tx^ z; y^ ±Xj z; 0H+ (x, ±y, z;

X, ±2/ J z;y^ ±x, z; y, ±x^z)}

Pj. / 111. 313 ) T F4^/ddm 16(c).. 2/m CTl 88 8j 888'!/

14^/amd 8(c).2/m. 1+ rO^gj 0^^; ^Oj; ^Oj)

. .2^T2x I4yamd(f) 1+ (iXj Uj 8 J iXj 5j 8j 5j —Xj 8j iXj s)

78 2 ) ;

. .2CIlz P4/nmm 2(o' )4rm 00z;\\z

.2^.CIlz2xx P42^m(e') lO+ (XXZT ..2CIlz'+xx P4/nrm(j') xxz;xxz;xxzt;\10+(xxzt;xxz;xxz)

4^. .P I Ix P4„22 4(a' ).2. xOO; 0x1; xOl; 0x1 3 cc c 3

4-. .P I lx2yz P4^22(d') xyz; xyz; 00i+(yxz; yxz ) ;00\+(xyz; xyz ) ; ^ ^ {5^.?P ^Dlxx2yz} 001+ (yxz; yxz)

4^..P ^Dlxx PU^22 3 cc 3

4^. . I ^Dlxx P432^2 4 (a).. 3 c lU+(xx0)

. I^^Dlxx2yz P4^2^2(b) xyz; yxz; 5 H+ Tyxz; xyz ) ;00\+ (xyz; yxz ) \\l+(yxz;xyz)

4..^TF Ix I42d 8(d). 2. -Z^+ (X X 83' c u Q y U'^' 8 J 8 )

. . 22^TC Ix 14^22 2. X+ (X L, Q y X Q y iiX Q y nX 8 ) cc 8(f).

^••Vc2l^ m^/acd 16(e').2. 1+ (xOO; x\ \; xO\; x\0; 1 1.3 3, 1.3 1, 1. I 3_ 11

^g.-P^^ Dl^lxy P4g 4(a')l xyO;yxl;xy\;yxl

Table 22: Orthorhombic System

P Pmmm l(a)mmm 000

P2x Pmmm(j) ±(x00)

P2y2z Pmmm(u) ±(0^y^±z) 2..P 2>Q^ Pccm(q) +(xyO) ;±(3:y\) . m. .P^2xz Prma(i) +(xOz) ;\00±(xOz) {?2.P^Blz2x} .m.Pgj^2yz Crma(m) C±(Oyz;\yz) {2..P Flz2y}

79 )

P2x2yz P222(u) xyz;xyz;xyz;xyz P2x2y2z

.mP 2x2yz Pcom (r' ) xyz;xyz; xyz; xyz; 00\+ (xyz; xyz; xyz; xyz) {5. .P 2xy2z} m. .P,2xz2y Pmma(l) ±(x^±y^z); iOO+(x^±y,z) {72. P Blz2x2y} ,22P 2xyz Poaa (f) ±(xyz);lOO±(xyz) ;00\±(xyz);lO\±(xyz) {c?2P Fly2xz} {12.(f^Blz) 2xy} ^ 2.inP 2xyz Pbcm(e) ±(xyz) ;OlO±(xyz) ;00\ + (xyz) ;Ol\+(xiyz) {.§jin(P Clx) 2yz} {2.1P^ A^C ^^lxy2z} be Td c m P^l-Sx^yz Crma(o') C+{xyz;xyz;xyz;xyz; lOO+(xyz;xyz;3yz;xyz) }

{ .m.P 2yz2x} {m. .P 2xz2y} {2. .P^Flz2x2y}

22 P22xyr Iboa(f) 1+ { ± (xyz ) ; loot (xyz); 0\0± (xyz );00\±(xyz)} {:22(P Fix) 2yz} {2.2(P^^Fly)^2xz} {22.(P^^Flz)^2xy} ab c

Cmmm 2(a)mm 000; l\0

C2x Cnvm(g) C± (xOO) C2z Cmmm(k) C±(00z)

C2y2z Cmrm(n) C±(0,y,±z) C2x2y Cmmm(p) C± (xj, ±y, 0) .2.B2yz Pmna(h) ±(0yz);\0\±(0yz) b. .C2xy Pbam(g) ±(xyO);UO±(xyO)

.n.C 2yz Cmam (f) C+{±(0yz);00l±(0yz)} {§. .C Fly2z} n. .C F2xy Ccom (I) C±(xyO;xy\) b. .C^2xy Ibam (j) I± (xyO;xy\) .2.B^2yz Imma(h) I+{±(0yz);0l0±(0yz)} {.2.B^A lz2y} b a -' C2x2yz C222(l) C+ (xyz;xyz;xyz;3yyz) C2x2y2z Cmrm(v) C±(x,y,±z;x,y_^_+z) b. .C2x2yz Pban(m) xyz;xyz;xyz;xyz;WO-¥ (xyz; xyz; xyz; xyz) 2.2A 2xyz Pnna (e) ±(xyz);0\\±(xiiz) ; hOO+ (xiiz J ; U ii/^ys; {?/n. .B A lz2xy} {. .2/aB^F5x2yz} .2.B2yz2x Prma(i) ±(±x,y,z);lO\±(±x,y^z) b. .C2xy2z Pbam(i) ±(x,y, ±z);\\0±(x:,y,±z)

b2,C^2xyz Pbcn (d) ±(xyz); \\0±(xyz) ;00\±(xyz) ; \\\±(xyz)

.maB^2xyz Pnma (d) ±(xyz); lO\±(xyz) ;0\0±(xyz) ;\ll±(xyz) {1.2,B A FI lxz2y} ^ .n.C 2yz2x ^ Cmcm(h) C+{±(±x,y,_z)_;iJl±(+x,y,z)_} n. .C^2x2yz Cccm (m' C+{xyz;xyz;xyz;xyz; m+(xyz; xyz; xyz; xyz) } {n. .C F2xy2z}

b. ."C 2x2yz Ibam(k^ ) 1+ {xyz; xyz; xyz; xyz; 1^0+ (xyz; xyz; xyz; xyz)} {%. .C 2xy2z}

80 ) ± )

,2.B 2yz2x ImmaCj) I+{±(±x,y,z);lO\±(±x^y,z)] {2. .A 2xz2y} {.2.B^A Iz2x2y} b a

Immm 2(a)mmm 000;\\\

I2x Irrmn(e) I± (xOO)

I2y2z Imrm(t) l±(0,y,±z)

n. . I2xy Pnnm(g) ±(xyO);\\\±(xyO)

I2x2yz 1222 (k) J+ (xyz; xyz; xyz; xyz I2x2y2z Immm(o) I±(x,y,±z;x,y,±z)

n. . I2x2yz Pnnn (m) xyz; xyz; xyz; xyz; 1^1+ (xyz; xyz; xyz; xyz )

n. • I2xy2z Pnnm(h) ±(x,y, ±z) ;\\\ + (x^y,±z)

Fmmm 4(a)mrrm 000; OUk

F2x Fmmm(g) F± (xOO)

F2y2z Fmrm(m) F±(0,y,±z) .2.F2yz Cmaa(f) C+{±(0yz);\0\±(0yz)}

F2x2yz F222(k) F+ (o^yz; xyz; xyz; xyz F2x2y2z Fmmm(p) F± CxjZ/j ±z;x,y^_+z) c.2F2xyz Pcon(e) ±(xyz);0\\±(xyz); \Ol+(xyz); l\0±(xyz) {12-L.(CIlz) 2xy} ^ bc.F2xyz Pbca(c) ±(xyz) ;0\l+(xyz); lOl±(xyz) ; HOtfxyz) .2.F2yz2x Cmca(g) C+{±(±x,y,_z)_;lOH(±x,y^z)}

c..F2x2yz Caaa(i) C+{xyz;xyz;xyz;xyz; \0\+(xyz;xyz;xyz;xyz) }

Fddd 8(a)222 F+(000;lU)

D2x Fddd (e) Ft { (xOO) ;lll± (xOO) }

d. .D2xy Fdd2(b') F^{±(xyO);Ul+(xyO)}

d, ,D2x2yz Fddd(h) F+ {xyz; xyz; xyz; xyz; Till+(xyz;xyz;o!yz;xyz) }

Vj. fill • 111^ ) Fddd 16(c)l TTl 88 8j 888''^/

.2.P Blz Pmma 2(e' )rm2 OOz; jOz 3.

.2.P^Blz2y Pmma(k ' o,±y,z; I:, ±y:,z

.2.P^Blx2yz P222^(e) xyz; xyz; 00\+ (xyz; xyz ) {2. .P^Aly2xz}

81 ) )

.2^. .CIlz Pmmn 2(a)mm2 00z;\\z

2^. .CIlz2y Pmmn(e) 0,^yyz;\,\±y,z

2 . .CIlz2xy P2^2 2(c) xyz;xys; ^lOi-(xyz;xyz)

.CIlz2x2y PmmnJcf •A'j ^ +7 2^, —y J '^'j J *^ J 2 2*-^ ' I —i7 J "J

2, . .C Fly Cmcm 4(c')m2m C+(OyO;Oyl) 1 c 2...C Fly2x Cmom(g' C+(±X:,y,0;±x,y,\) 1 c

.2 .C Flx2yz C222^(o) C+{xyz;xyz; l^l+(xyz;xyz)} (5 C Flv2xz) 1 c

2. .P ^Flz Cmma 4(g')rrm2 C+(00z;0\z) ab

.2.B^A Iz Imma 4(e')rm2 I+(00z;0\z) b a

. .2C^Bjjlx2yz 12 2 2Jd') I+{xyz;xyz; \\0+ (xyz;xyz)} {2.. A C ly2xz} {.2.BfA'^lz2xy} D a

1-77 1 Pbcm 4(d')..m xyO;xy\;x, \^y,0;Xy 2 y 3 2

l—y. 1 1 1.2,B, A FI Ixz Pnma 4(c').m. XOZ S 2 "^X 0 't'j +2 1 b a a J 2 J y ^ 3 2 2J 2

2, 2,. FA B.C I LI Ixyz P2,2,2, 4(a)l xyz^ \~X3y \+zj \+x, 3; 11 a b c a D c ^ 111 3 Xj i+y } 2~z

Table 23: Monoclinic System

P2/m 1(a) 2/m

P2y P2/m(i) ±(OyO)

P2xz P2/m(m) ±(xOz)

P2xz2y P2/m(o) ±(X3 ty^z) mPj32xyz P2^/m(f) ±(xyz);OlO±(xyz) {2^Pj^ACIlxz2y} 2P 2xyz P2/a(g) ±(xyz) ;00l±(xyz) {cP Aly2xz}

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

C2y C2/m(g) C±(OyO)

C2xz C2/m(i) C±(xOz)

C2xz2y C2/m(j) C±(x,±y,z) cA2xyz P2 /c(e) ±(xyz) ;Ol^±(xyz) 2 CcF2xyz C27o(f) C+{±(xyz) ;0\\±(xyz)} {lC^Fly2xz}

cP Aly P2/c 2(e')2 OyO;Oy\ c

IC Fly C2/c 4(e ')2 C+(OyO;Oyl) c ^

2,P^ACIlxz P2^/m 2(e')m xOz;x\z 1 D

Table 24: Anorthic System

P 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

1 1 1 mZm Pm3m(a), (b)\\\ 42m P43m(a)j (b)lH

432 P432(a) , (b)\\\

m3. Pm3 (a), (b)\\\

23. P23 (a)j, (b)\l\

8 43m Fm2m(c)lll ( ) P2 432 Fm2a(a)

m3. Fm3c(b)

23. F43c(a), (b)Hl F432(c)lU

-T-l >-? / 1111 Fm3 [G)ll\

-r- n / » 1 1 1 . 2m Im3m(o) lii

. 32 Pm3n(e) HI 1432(a) 5U

.3. Pn3n (a) Im3 (a)lH Ia2 (a)^ (b)lll

1 1 2 m2m Im2m(a) 43m Pn3m(a)

Td ^n )

432 'Pn3n(a) 1432(a)

m3. Pm3n ( a)

Im3 (a)

23. P43n(a) P4^32(a)

Pn3 (a)

123 (a)

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

85 . m ,.

(^) I2 23. Fd3o(a)

• 3.

0 3 4/rm.m Pm3m(a), (d)\\\ 42. P43m(o), (d)U{ 42.2 P432(a), (d)Ui mrm. Pm3 (c)Jd)lV2 222. P23 (c),(d)m

(^) J2 24 4m. 2 Fm3c(c)lU 4/m.. Fm3o(d)

4. F43o(g), (d)lU m.rm Fm3m(d)

2. 22 F4 32(d)

2/m. FmS (aj

4 m3m Fm3m(a), (b)\ll 43m F43m(a), (b)Ul, (c)Uh (dJUi 432 F432(a), (b)\\l m3. Fm3 (a)Jb)V2i 23. F23 (a), (b)n\, (o)Hh (d)m

. 3m Pn3m(h)Hh (o)Ui .32 P4^32(b)lU, (c)lU

.3. Pn3 (b)Hh(<2)Ui Pa3 (a),(b)\\\

32 .32 Fd3o(b)lH

. 0. CCiOa ( 8 8 8

4 .32 P4^32(d), (b)Ul

) Y 4 . 32 P4 32(a) (b)^^

6 4m. 2 Pm3n(c), (d)\\\

4. P~43n(e)\\\, (d)

j: Ct \ t:^ 2. 22 P4^ ^032(e)/ y (f)\\\\ J y 2 2 2

6 4/mm.m Im3m(b)

42. m Pn3m(d)

(^)200/020/002. (^) 100/010/001. .

86 . .

I43m(h)

42. 2 Pn3n (b) 1432(b)

mmm. Pm3n (b)

ImS (b)

222. . P43n(b) P4^32(d) 6 Pn3 (d)

123 (b)

48 4. . Fd3c(d)

0 8 43m Fd3m(a), (b)in 22. F4^32(a), (b)\l\ Fd3 (a),(b)lH

8 .32 14 ^32 (a)

~Y# 8 .32 I4^32(b)

12 4m. 2 Im3m (d)

4. Pn3n(d) I43m(d) 2.22 Pn2m(f) 1432(d)

El 12 4. . I43d(a)

's 12 4. . I43d(b)

12 2.22 14 ^32 (a)

O 0 0 T/l 7 9 ) (^) ~V 12 14 jOii ( CLJ

B 16 . 3m Fd3m(c), (d)V2\ .32 F4^32(c), (d)\i\

.3. Fd3 (c),(d)\\\

16 .32 Ia3d(b)

(^)200/020/002, () 100/0 10/001.

87 . .

24 4. . Ia3d(d)

24 2. 22 IaZd(a)

Cubic, univariant

P4xxx 4 . 3m P43m(e)

.3. P23(e)

(^) (P Hxxx ) 32 .3. Fl3c(e) 2

2^2^. .FYlxxx 4 . 3. P2^3(a)

P6z 6 4m. m Pm3m(e), (fJli'z

4. . P432(e), (f)\\\ mm2. Pm3 (e)Jh)\\l

2m mm P43m(f) . (a)iH

2. . P23 (f),(i)l\l

(^) (P6z)2 48 4. Fm3c(f)lU mm2. Fm3c(e)

2. mm Fm3m(g)Hi 2.. Fl3a(f), (g)lH F432(i)lll Fm3 (g)lH

.3.J2x 6 mm2. Pm3 (f),(g)\\\ 2.. P23 (g),(h)m

P8xxx 8 . 3m Pm3m(g)

.3. P432(g)

Pm3 (i)

(^) (P Sxxx)^ 64 .3. Fm3o(g)

. . 2mxxx 8 . 3m Pn3m(e)

. 3. P4^32(g)

(^)200/020/002.

88 ...

Pn3 (e)

(^) (..2I4xxx)2 54 . Z. FdScie)

I4xxx 8 . 2m I43m(o)

.3. P43n(e)

123 (a)

.'''Y2XXX 8 .3. P4^32(a)

0 r) '4^..~Y2xxx CI . o.

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

Q •2 T9 7.f/-r) 2^2^. .P2Y^flxxx o . c. J.CI \ a J

. 3. J4x 12 Tnm2. Pm3m(h)

2.. P43m(h) P432(n)

P12xx 12 m, m2 Pm3m(i), fjiHi ..2 P432('C)^

(^) (I=12xx)2 96 ..2 Fm3o(h)iH

.3.W2Z 12 mm2. Pm3n(g),(h)\V2

2.. P43n(g)\Ui (h) P4^2{%) ,{3)1^^

I6z 12 4m.m Im3m(e)

4.. Pn3n(e) 1432(e)

rm2. PmZn (f) Im3 (d)

2. mm Pn3m(g) I43m(e)

2.. P43n(f) P4^32(h)

Pn3 (f)

(^)200/020/002. (^) 100/010/001.

89 . )))

12 S rdj

9

14^. .F3xx 22 . . 2 1P4 ^32 (k) Ti'ii'ii* ( 1^ ) Ti 6

[4 /Y3xi< 22 . .2 P4^32(d)

&) "Y3xx i2 . . 2 P4^32(d) 1

.3.J^2x 22 m?772. . Im3 (e)

2. PnZ (g)

160 \ &

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

1 8 XXX 16 . Sm ImSm ( f)

. 3. Pn3n (f) PmSnd)

1432(f)

Im3 (f)

FM-xxx 16 . 3m F43m(e)

m tJ » F23 (e)

TA 7/7 4. . I^Yff-jflxxx lb

22. . Y*2xxx 16 . 3. I4^32(e)

22. .P22XXX J. 0 Tn3 (r

PrnZyi ' *^ ( 1 . 3. WHxx 24 . . 2 Lilt KJ ^ fj /

5. . F6xx 24 . . 2 Pn3m(i)Hh (j)iU

F6z 24 4m. m Fm3m(e)

4. . F432(e)

mm2. . Fm3 (e)

-T-i n t-> / r% \ / 1111 2. mm F43m(f), (g)Hl

(^)200/020/002. (^) 100/010/001,

90 .

2. . F22 (f)Jg)Ul

. 3 K 24 mm2. . Im2m(g)

2. . Pn2n(g) Pn2m(h) I42m(f) 1422(g)

1 12 XX 24 m. m2 Im2m(h) ..2 Pn2n (h) 1422(h)

.3.S2Z 24 2. . I42d(d)

4. .P^Sxx 24 ..2 I422(i)lll

.3.V2Z 24 2. . 14^22 (f)

4^. ."'"Y*3xx 24 ' 2 I4^22(h)

( ) 4^. . Y*-3xx 24 . . 2 I4^22(g)

.3.J2S^fV*lx 24 2. . Ia2 (d)

F8xxx 22 . 2m Fm2m(f)

. 2. F4 22(f)

Fm2 (f)

. .2D4XXX 22 . 2m Fd2m(e)

. 2. F4^22(e)

Fd2 (e)

4. . Y**2xxx 22 . 2. Ia2d(e)

F12XX 48 m.m2 Fm2m(h), (i)l\\ ..2 F422(g), (h)\\\

(^) 100/010/001

91

501-563 0 - 73 -7 ..

D6z 48 2. mm Fd3m(f)

2. F4^32(f)

FdZ (f)

4. .P^exx 48 ..2 ImZmd) III

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

^a. . Yjf*3xx 48 ..2 Ia3d(g)

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

4. .T5xx 96 ..2 Fd3m(h)

42, .F^Sxx 96 ..2 Fd3o(g)U\

Cubic, bivariant

P6z2xx 12 . . m P43m(i)

P6z2x 12 m. Pm3 (3)Jk)l\\

P6z4x 24 m. Pm3m(k),

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 m. . ImZm (j )

I5z4xx 48 . .m Im2m(k)

F6z2xx 48 . . m F42m(h)

F5z2x 48 m. Fm3 (h)

F6z4x 96 m. FmZm (j)

F6z4xx 96 . . m Fm2m(k)

. .2P26Z2X 96 m. FmZad)

. .2D5z2xx 96 . . m FdZm(g)

Cubic, tri iant (site symmetry 1)

The 12-pointers P6z2x2y Pm3 (I) P6z2xy P23 (3)

(h) n. . I6z2xy Pn3 2^2^. . FYlxxx3yz P2^3(b)

Pa3 (d) The 2M--pointers be. . F5xyz P6z2xx2y P43m(j) I6z2xy 123 (f)

. . cI6z2xy P43n(i) 2^2^. .P2Y*#lxxx3yz 12^3(0)

P6z4xy P432(k) The M-S-pointers P5z4x2y Pm3m(n) . .2I6z2xy P4^32(m)

Pn3n(i) n. . I6zM-xy 4 . . Y3xx2yz P4^32(e) —h ( ) H . . Y3xx2yz P4^32(e)

. .2I6z2x2y Pm3n(l)

(^)ioo/oio/ooi.

93 ;

. .2I6z2xx2y PnZm(l) ,2D6z2xx2y Fd3m(i)

I5z2xx2y I43m(h) d.2l26z2xy Fd3o(h)

, 3dS4xyz I4Sd(e)

I5z4xy I432(j)

22. . YJf3xx2yz

I5z2x2y Im3 (h)

22. .P^Bxyz Ia3 (e)

F6z2xy F23 (h)

I5z4x2y Im3m(l)

43. . Y*)f3xx2yz Ia3d(h)

The 96-pointers: F6z2xx2y F43m(i)

,nP25z2xy F43a(h)

F6zM-xy F432(q)

. .2D6z2xy F4^32 (h)

F6z2x2y Fm3 (i)

d. .D6z2xy Fd3 (g)

The 192-pointers F5z4x2y Fm3m(l)

m. . P^Bz^jQ^ Fm3o

94 . .. .

Occurrence of Lattice Complexes Table 26-27 Lattice complex Mill ? oixe uccurrence in all its plicity symmetry space group, Wyckoff letter, representations (oriented) shift from standard representation

Hexagonal and rhombohedral , invariant

0 6/rrmm P6/mmm(a)y (b)OOl 6m2 P6m2(a), (b)OOl, (c)\\0, (d)\n, (e)\\0,(f)n\ 62m P62m(a), (h)00\

622 P622(a)j, (b)00\

6/m. P6/m(a), (b)OOl

6.. P6 (a), (b)OOl, (c) \\0, (d) Wz, (e)nO,(f)m

3m. P2ml(a), (b)00\ 3.m P21m(a)^ (b)OOl

22. P221(a)^ (b)00\

2.2 P212(a), (b)OOl, {e}\\0, {d)\\\,

2.. P2 (a), (b)00\

P[z] 6mm P6mm(a)

6. P6 (a)

2m. P2ml(a), (b)ilO, (a)\\0 2.m P21m(a)

2. P2 (a)Jb)\\0,(c)%\0 (^) 6m2 P6jmmc(b)00l 62m P6 ^/mom(a)00l 622 P6/moQ(a)00l

6/m. P6/mao(b)

6. P6a2(b)00l, (d)\U, (f)m P62c(b)00l P6^/m(a)00l

3m. P6^/mmo(a)

'2.m P6^/mcm(h)

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

95 ..

32. P62c(a) P6^22(a) P3cl(a)00l Z.2 P6c2(a), (c)HO, (e)ViO P6^22(b)00i PZlc(a)00i

3.. P6^/m(b) PZcl(b) PZlc(b)

2 6. . P6cc (a)

3m. P6^o(a) Z.m P6 ^cm(a)

3. P6^ (a) PZal(a), (b)\iO, (c)no PZlc(a)

(^) Z 222 P6^22(a), (b)OOl P6^22(a), (b)OOl

(^) P^W Z 2. . P6^ (a)

2 6m2 P6/mrm(o), (d)OOl 0I. J 6.. P62m(a), (d)OOl

P6/m (a), (d)OOl

Z. 2 P622(o), (d)OOl PZlm(c), (d)00\

G[z] 3m. P6mm(b)

3. P6 (b) PZlm(b)

(^) G 4 6. P6/mco(d) c P6^/mcm(o)00l Z.2 P6/mco(c)00l P6^/mcm(d)

(^) G [z] 4 Z. . P6ca(b)

irO 2 ^ '

1 HI1 2 6m2 P6jmmo(o), (d)00\ (^)100/010/002, (^)100/010/003.

96 ... 1

6. P62o(c), (d)00\ P6jm(c), (d)00\ 3 3. 2 P6^22(o), (d)00\ 0 P31c(c), (d)00\

E [z] 2 3m. P6^mo(h)

3. . P6^ (b) P31a(b)

B 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)

'R 6 32. R3o (a) 00 c

3. . R3c (b)

(^) 6 3. R3a (a)

3 mmrn P6/mm(f), (g)00\ i_j 222 P622(f), (g)OOl

2/m. . P6/m(f), (g)00\

.2/m. P3ml (e), (f)00\ ..2/m P31m(f), (g)00\

1 P3 (e),(f)00h

N fz] 3 2mm P6mm(c)

2. P6 (c)

N 6 222 P6/mco (f)OOl c

2/m. . P6/mca(g)

. 2/m. P6 ^/mma(g) 6

. . 2/m P6Jmom{f) 6 1 P6jm(g) 6 P3cl (e) P31o(g)

(^) NJz] 6 2. . P6co(c)

3 222 P6^22(a), {d)00\ 2

+Q[zJ 3 2. . P6^ (b)

(^) -Q 3 222 P6^22(c), (d)00\

(^)100/010/002. ('^)010/110/002. ('^)100/010/001.

97 .

(^) 3 2. . PS, (b)

(e) [71 9 . 2/m. H3m (d)00\,

1 R3 (d)OOl, (e)

'M 18 1 R3g (d) c

Hexagonal and rhombohedral , univariant

P2z 6mm P6/mrm(e)

6.. P622(e) P6/m(e)

3m. P6m2(g)Jh)\lO, (i)\\0

P3ml (a) 3.m P62m(e) P31m(e)

3.. P6 (g),(h)\\0. ii)\\0 P321(c) P312(g), (h)\iO, (i)l\0

P3 (c) P6/moc(e) ( ) (P2z), 6. .

3m. P6^/mma(e) 3.m P6^/mcm(e)

3. P6c2(g),(h)\\0, ii)\\0 P62o(e) P6^22(e) P6^/m(e)

P3al (c) P31o(e) P6^22(e) r) (P2z), 2. . P6^22(e)

.2.GE1Z 3m. P3ml (d)

3.. P321 (d)

P3 (d)

100/010/001 (^) 100/010/002. (^) 100/010/003. ('')010/110/002 . C'^)

98 .. )

(^) (.2.GElz) 4 3. P3al (d) c

P3xx 3 rm2 P6m2(j), (k)OOl ..2 P312(j), (k)OOi

P3xx£z} 3 . m. P3ml (d)

a 9 ( (^) (P3xx) u -i c

P3x 3 m2m P62m(f), (g)00\

.2. P321(e), (f)00\

P3x[z] 3 . . m P31m(c)

(^) (P3x) 0 9 c

32..P(."^Qlxx 3 ..2 P3^12(a)00i, (b)OOl (d) 3^..P^ Qlxx 3 . . 2 P3^12(a)00\, (b)OOl

32..Pc^Qlx 3 .2. P3^21(a)00i, (b)OOi

9 i^) 3^..P^"q1x • ^ • P3 21(a)00\. (h)OOi

G2z 4 3m. P6/mmm(h)

3. . P62m(h) P622(h) P6/m(h) P31m(h)

(^) (G2z)2 8 3. - P6/mao (h)

Jrb ^/mom{ nj

E2z 4 3m. P6^/mmc (f)

3. P62o(f) P6^22(f) P6^/m(f)

P?7^ Z' -P) ro 1(J { J J

R2z 6 3m. R3m (o)

3. . R32 (a)

R3 (c)

•(R2z) 12 3. . R3o (c) c

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

99 .

N2z 2rm P6/mrnm(i)

2. P622(i) P6/m(i)

( ) (N2z) 12 P6/mco(i)

P5x m2m P6/rrmn(j), (k)00\

.2. P622(j), (k)OOl Pdml(g), (h)OOl

P6x [z] 6 . .m P6rm(d)

(^) (P6x) 12 P6/mcc (3)001 P6^/mmc(i)

P6xx rm2 P6/mrm(l), (m)00\ ..2 P622(l), (m)OOl

PZlmd), (c)00\

P6xx [z] 6 . m. P6mm(e) (^) (P6xx) 12 ..2 P6/moc(k)00l P6^/mcm(i)

2P 3x m2m P6 /mom(g)00i c

. 2. P6^22(g)

P'Zcl(f)00l

,2? 3x[z] . , m P6^om(c) c

, 2.P E3xx mm2 P6 Jmino(h)00l c ..2 P6^22(h)00l P31o(h)00l

,2.P E3xx [z] . m. P6^mc(o) c

2. P6 222(a)

(d) 32.2P^^-Q^lx 2. P6^22(a)

..2 P6^22(b)00'^ ..2 P6^22(b)OOh.

Q2z 6 2. P6^22(f)

(d) Q2z 6 2. P6^22(f)

(^)100/010/002. (d)100/010/001.

100 .,

32..P^'^Q2x 6 .2. P6^22(g), (h)OOl (d) 3^..P^-(32x 6 . 2. P6^22(g)j (h)00\

32 - .Pp''"Q2xx 6 ..2 P6^22(i)00\, (j)OOi (d) 3^..P^ (32 XX 6 . . 2 P6^22(zJ00ij (3)001

R3xx 9 , m. R3m (b)

R3x 9 .2. R32 (d),(e)00\

R6x 18 .2. R3m (f),(g)00\

.n. 'R 3x 18 .2. R3c (e)OOl c

Hexagonal and rhombohedral, bivariant

P3xy 3 m, P6 (0), (k)OOl

P3xy fz] 3 1 P3 (d)

32..P^'^Qlxy [z] 3 1 P32 (a) (d) 3^. .P^"Qlxy[z] 3 1 P3^ (a)

P3xx2y 6 m. . P6m2(l), (m)OOl

P3}

P3xx2z 6 . m. P6m2(n)

. . 2P 3xy 6 m. P'Bo2(k)00l c

. .2P3xy [z] 6 1 P3g1 (d)

P3x2z 6 . , m P62m(i)

P3x2y 6 m. . P62m(o)Jk)00\

P3 K2y [z] 6 1 P31m(d)

(^)100/010/001.

101 ..

.2.P^E3xy 6 m. . P62c(h)00l

.2.p^e;3xy [z] 6 1 PZlc(c)

P6xy 6 m. . P6/m(j), (k)00\

P 6xy {zl 6 1 P6 (d)

2, . .P ESxy 6 m. P6^/m(h)00l 1 c

2^ . .P E3xy{zl 6 1 P6 ^ (a) 1 c

P6xz 6 . . m P31m(k)

P6xxz 6 . m. PZml (i)

6 1 P6^ (a)

i)6 1 P6 ^ (a)

32..P^^Q2x3r tz) 6 1 P6^ (a)

(d) 3^..P^-Q2xy{z] 6 1 P6^ (c)

R3xy iz] 9 1 R3 (h)

P6x2z 12 . . m P6/mmm(n)

P6xx2z 12 . m. P6/mmm(o)

P6x2y 12 m. . P6/rmm(p), (q)OOl

P6 x2y[z] 12 1 P6mm(f)

. 2.P^6xy 12 m. P6/moc(l)

.2.P 6xy{z} 12 1 P6co(d) c

. .2P^3x2y 12 m. P6^/mam(j)00l

3- . .2P <2y{z] 12 1 c

. .2P 3x2z 12 . . m P6^/mGm(k)00l c

(^)100/010/001.

102 .2.P E3xx2y 12 m. . P6^/mma(j)00i c .2.P E3xx2y(zl 12 1 P6^mo(d) c

.2.P E3xx2z 12 . m. P6 Jrmo(k)00l c o

R6xxz 18 . m. R3m (h)

R3xx2y [z) 18 1 RZm (a)

.n. ' R 3xy [z] 18 1 R3c (b) c

Hexagonal and rhombohedral , trivariant (site symmetry 1)

The 6-pointers:

P3xy2z P6 (I) .2.P^E3xy2z P62Q(i)00l

P3xx2yz P312(l) P6x2yz P622(n)

P3x2yz P321 (g) 3r2Pcc\^^2yz P6^22(o) (d) " 3^.2P_ Q lx2yz P6^22(o) z LC c 32..P^^Qlxx2yz P3^12(c)OOi

C^) 3^. .P^-Qlxx2yz P3^12(o)00\ 32- .P^^Q2x2yz P6^22(k) (d) 3^..P^-Q2x2yz P6^22(k) 32..P^'^Qlx2yz P3^21(c)OOi

{^) 3^. .P^-Qlx2yz P3^21(c)00\ . .2P 3x2yz P6^22(i) c

P6xyz P3 (g) P6xy2z P6/m(l)

The 12-pointers: 2, . .P E3xy2z P6^/m(i)00l 1 c P3xx2y2z P6m2 (o) P6xz2y P31m(l) ..2P^3xy2z P6g2(1)001

. .cP^E3xx2yz P31a(i)00l P3x2y2z P62m(l) (d)lOO/010/001.

103 :

P6xxz2y P2ml (j)

P?,nl (n)OOT- c

The 18-pointers:

R3x2yz R32 (f)

R6xyz R3 (f)

The 24--pointers P6x2y2z P6/mrnm(r)

m. . P 6x2yz P6/mca (m)OOl c

. .2P 3x2y2z P6^/mam(l)00l c

.2.P^E3xx2y2z P6jmmc(l)00l O

The 36-pointers:

R6xxz2y RZm (i)

•n. ' R 3x2yz RZo (f)OOi c

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

0 1 4/mmm P4/mrm(a), (b)00\, (c)\\0, (d)\\l 42m P42m(a), (b)\H, (c)00\, (d) \\0 4m2 P4m2(a), (h)l\0, (c)Uh (d)00\ 422 P422(a), (b)00\, {c)\\0, (d)\\\

4/m. P4M(a), (b)OOi, (o)\\0, (d)\V2

4. . P4 (a)Jh)00\,(o)\\OJd)\\\

P{z} 1 4mm P4rm(a), (b)llO

4. . P4 (a),(b)\\0

(^) P 2 42m P4^/mom(a)00l, (b)\\l c 4m2 P4^/mmc(e)00l, (f)\\l 422 P4/mcc(a)00l, (c)\\l

4/m. . P4/maa(b), (dJhlO

4. . P42a(e), (f)\\0 P4c2(c), (d)\lO P4^/m(e)00l,(f)\\l mmm. P4^/nmo(a), (b)V20

m. mm P4^/mom(b), (d)\\0

222. P42c(a)00l, (c)\\l P4^22(a), (b)\\0

2. 22 P4G2(a)00l, (b)\\l P4^22(e)00l, (f)\U

2/m. . P4^/m(a), (b)V20

2 4. . P4ac(a), (b)\\0 2mm. P4^mc(a), (b)llO

2. mm P4^am(a), (b)\\0

2. P4^ (a), (b)UO

&) c 2 42m P4/nbm(c)0\0, (d)0\\ 4m2 P4/nxm(a), (b)00\

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

105 .

422 P4/nbm(a), (b)OOl

4/m. . P4/mbm(a), (b)OOl

4. . P42jm(a), (b)00\

P4b2(a)j, (b)00\ P4/n(a), (b)00\

mm. P4/mmm(e)0\\j (f)0\0 m. mm P4/mbm(o)0\l, (d)0\0

222. P42m(e)lOO, (f)\0\ P422(e)\OOj, (f)\0\

2. 22 P4b2(o)OiO, (d)0\\ P42^2(a)j (b)00\

2/m. . P4/m(e)0\0, (f)Oll

&) C(2) 4. . P4bm(a)

2mm. P4mm(o)\00

2. mm P4bm(b)\00

2. . P4 (a)0\0

. . 2/m P4/nbm(e)U0, ab (f)Ul P4/nmm(d)llO, (e)Ul

1 P4/n(d)U0, (e)ll\ i') C 42m I4/mcm(b)0\l 4m2 I4/mmm(d)0ll 422 I4/mcm(a)00l

4/m. I4/mcm(o)

4. . P4/nnc(d)l0l

P4/noc (b) P4^/nbo(d) P4^/mba(b)00l P4^/mnm(d)0U P4^/nam(b) I4a2(b), (o)0U I42m(d)0\l I4/m(d)0U mmm. I4/mmm(a)OlO

m. mm I4/mam(d)0\0

222. P4/mco(e)0\l P4/nnc(c)\00

( )110/110/001. (^)200/020/001. () 110/110/002.

106 P4^/mcm(e)0\l P4^/nho(a)0U^ (b)OOl P4^/nnm(a)OlO I42m(o)0\0 I422(o)0\0 2.22 P4/mna(d)0U P4/nac(a)00l P4^/nho(o)0\0 P4^/nnm(d)0\l P4^/rribc(d)0U P4^/ncm(a)00l I4G2(a)00l,(d)0\0 1422 (d)OU

2/m. . P4/moc(f)0\0 P4/mnc(c)OlO P4^/mom(f)0\0 P4^/rrbo(a), (c)0\0 P4^/wnm(o)0\0 I4/m(c)OiO

4.. I4om(a)

2rnm. I4rm(b)OlO

2. mm I4cm(b)\00

2.. P4^om(o)0\0 P4^nm(b)0\0 P4oc(q)010 P4no(b)0\0 P4^bo(a), (b)OlO 14 (b)0\0 ..2/m I4/mrm(f)lU I4/mom(e) HI

1 P4/nnc(f)Ul P4/naa(d)H0 P4jnbo(e)lU P4^/rmo(e)lll I4/m(f)\ll

(^)110/110/002. (®)200/020/002.

107*

501-563 O - 73 - 8 4/mmm I4/mmm(a), (h)00\

42m P4^/nnm(a), (b)OOl I42m(a), (h)00\ 4m2 P4^/nma(a), (b)00\ I4m2(a), (h)OOl, (c)0\l, (d)OU 422 P4/nnc(a), (b)00\ 1422(a), (b)OOl

4/m. . P4/mnG(a), (b)00\ I4/m(a), (b)00\ P42^c(a), (b)OOl P4n2(a), (b)00\ P4^/n(a), (b)OOl 14 (a),(b)00l,(c)0\l,(d)0\l mmm. P4^/nma(c)0\0, (d)0\\

m. mm P4^/mnm(a), (b)00\

222. P42c(b)\0l, (d)0\l P4^22(o)0\0, (d)On

3 2.22 P4n2(c)0U, (d)Ol' 2 u P4^2^2(a), (b)00\

2/m. . P4^/m(c)0\0, (d)OH I[z] 4mm I4mm(a)

4. . P4nc (a)

14 (a)

2mm. P4^mc(G)0\0

2. mm P4 ^nm(a)

2. . P4^ (c)OlO

F 135 . . 2/m P4^/nnm(e)lll, (f) P4^/ncm(a)Uh (d)Ui

1 P4^/n(o)lU, (d)m (^) F 4. . I4^/aad(b)

2. 22 I4^/acd(a)00l

( ) F fz) 2. . I4^od(a) c (^) I2 16 1 I4^/aad(c)0U

(^)110/110/001. (^)110/110/002. ( )200/020/002.

108 ...

8 42m F4 Jddrn- [3 1 (^) 4 4m 2 I4^/amd(a), (b)00\

4. I42d(a), (b)00\

14^/a (a), (b)00\

2.22 14 ^22 (a), (b)00\

4 2mm. I4^md(a)

2. . 14^ (a)

16 . . 2/m F4^/ddm^

8 . 2/m. I4^/amd(c), (d)00\

1 I4ja(o),' (d)OOl 1

Tetragonal, univariant

P2z 2 4mm P4/mmm(g), (h)UO

4. P422(g), (h)^{0 P4/m(g), (h)llO

2mm. P4m2(e), (f)l\0

2. mm P42m(g), (h)llO

2. . P4 (e)Jf)\lO (^) (P2z) 4 4. . P4/mco(g), (h)UO n 2mm. P4^/mmc(g), (h)UO

2. mm P4^/mom(g), (h)\\0

2.. P42o(k), (l)\\0 P4c2(g),(h)\\0 P4^22(g), (h)\lO P4^/m(g),(h)nO

C2z 4 4. P4/nbm(g) P4/mbm(e)

2mm. P4/mmm(i)OlO P4/nmm(f)

2. mm P4/nbm(h)OlO

P4/rribm(f)0\0

Unconventional setting. Cp. next entry. (^)100/010/002. (b)110/110 001. (f )lfo/Ho/ooi.

109 . . .

2.. P42m(m)OiO P42^m(d)

P4b2(e), (f)OlO P422(i)0\0 P42^2(d) P4/m(i)OlO

• P4/n(f)

r) (C2z ) 8 4. . I4/mGm(f) c 2mm. I4/mmm(g)0\0

2. mm I4/mam(g)0i0

2. P4/mca (i)0\0 P4/nno(g)\00 P4/mno(f)0\0 P4/nao(e) P4^/mcm(k)OlO

P4^/nba(f)OlO, (g) P4^/nnm(h)0\0

P4^/rribc(e), (f)0\0 P4^/mnm(h)OlO P4^/ncm(f) I4c2(f), (g)0\0 I42m(h)0\0 I422(f)0\0 I4/m(g)0\0

. .2CI1Z 2 4mm P4/rmn(c)0\0

4.. ^42^2(0)010 P4/n(G)0\0

2mm, P4m2(g)0l0

2. mm P42^m(c)0\0

2.. P4 (g)0\0

(^) (..2CIlz)^ 4 4. . P4/noc {o)0\0

2mm. P4^/nmo{d)0\0

2. mm P4^/ncm(e)0\0

2. V42^(d)0\0

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

110 P4a2 (i)OlO M^2^2(d)0\0 P4^/n(e)0\0

P4xx m. 2m P4/mrm(j)^ (k)00\

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

Pi|xx{z] . .m P4rm(d)

(^) (P4xx) ..2 P4/moc(3 )00l P4^/rmo(n)00l

(^) C4x .2. P4/nbm(k), (l)00\

{^) (CUx) 16 .2. I4/mem(j)00i

P4x m2m. P4/mmm(l), (m)OOl, (n)l\0, (o)\\\

.2. P42m(i), (k)00\, (l)\\0 P422(l), (m)\\h (n)00\, (oJHO Pi+x[z} . m. P4rm(e), (f)\\0 (^) (PHx) .2. P4/mco(k)00l, (DWl 111 P4^/mcm{l)00l, (m) 2 2 U r) C4xx P4/nbm(i), (j)OOl P4/nrm(g), (h)00\

( ) (C4xx) 16 ..2 I4/rrmm(k)0ll I4/mam('L)00l

.b.C2xx m. 2m P4/mbm(g)0\0, ('h)0\\ ..2 P4b2(g)0\0, (h)Ol\

P42^2(e), (f)00\

.b.C2xx{z) . .m P4hm(e)0\0 (^) (.b.C2xx) ..2 P4/mnc(g)0\l

1 ^ P4^/mbc(g)0 2 t

.2P I2x m2m. P4^/mme(j), (l)00\, c (k)Uh MWO .2. P42c(g)OOl, (h)\\l, (iJVii, (j)OOl P4^22(j), (k)^U, (l)00\, (m)\\0

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

111 .

..2P I2x{z) .m. P4^mo(d), (e)liO

. c 3 ( ) .2.C F2xx P4^/nnm(k)0U, (l)0\ 14 c 2 P4jncm(g)00l, (h)OOl

,2.P 2xx m. 2m P4^/mom(i), (j)OOh c ..2 P4c2(e)00l, (f)OOl P4^22(n)00h, (o)OOl

.2.P 2xx(z) . . m P4^cm(d) . c r) ..2C 2x .2. P4^/nba(h)00l, (i)OOi c

.n. I2xx m. 2m P4^/mnm(f), (g)\\i ..2 P4n2(f)m, (g)OU

P4^2^2(e), (f)00\

,n. I2xx{z) . . m P4^nm(o)

I2z 4mm I4/mmm(e)

4. P4/nnG (e) P4/mna(e) 1422(e) I4/m(e)

2mm. P4^/mmo(i)OlO P4^/nmc(c) I4m2(e), (f)OU

2. mm P4^/nnm (g) P4 ^/mnm(e) I42m(e)

2. P42c(m)0U P42^a(o)

P4n2(e), (h)0\l P4^22(i)0\0 P4^2^2(o) P4^/m(i)0\0 P4^/n(f) 14 (e), (f)OH {^) (F2z) 16 I4^/acd(d)

(^)110/110/001. (^)110/110/002.

112 i+^..P I Ix 4 .2. P4^22(a)00l, (h)\U 3 cc c

(g) 4, . .? I ^Ix 4 . 2. P4^22(a)00i, (h)\\i 1 cc

H^..? "^Dlxx 4 ..2 P4^22(c)00l 3 cc

Dlxx 4 . . 2 P4 ^22(o)00i 1 cc

H^. .1 ^Dlxx 4 ..2 P4^2^2(a)

(g) ""d Lxx 4 ..2 P4^2^2(a) 1 c mxx 8 m. 2m I4/mrm(h) ..2 P4/nnc(k) P4^/nma(f) I4m2(g), (h)OU 1422(g)

I 4xx{z} 8 . . m I4mm(G) lUx 8 m2m. I4/mrm(i), (3)\lO 1 .2. P4/nnc(i), (j)OOl P4^/nnm(i), (j)00\ I42m(f), (g)00\ 1422(h), (i)00\

mx(z} 8 . m. I4rm(d)

.b.C 2xx 8 m. 2m I4/mom(h)0\0 c ..2 P4/noc(f)00l P4^/nhe(o)0\0 • I4o2(e)00l, (h)OlO 1422(3 )0H

.b.C 2xx [z), 8 • .m I4Gm(c)\00 c

V D2z 8 2mm. I4^/amd(e)

2.. I42d(o) I4^22(c) I4^/a(e)

U.7tF Ix 8 .2. I42d(d) c (^)100/010/001.

113 .

. .d D2x [z) 8 . m. I4^md(h)

.2.^D2xx 8 ..2 I4^22(d)

(^) .2.^ D2xx 8 ..2 I4^22(e)

..22^TC Ix 8 . 2. I4^22(f) cc

. .2^T2x 16 . 2. I4^/amd(f)

V D4xx lb . . 2 14 ^/amaig)

0 1 0 1^ j/CZOCX{ e J Ui;-§

.2.F 2xx 1 0 ? T4 /nr>rl ( F)nn^ c

Tetragonal, bivariant

PUxxz 4 . .m P42m(n)

.2^.CIlz2xx 4 . . m P42^m(e)0\0

P4xz 4 .m. P4m2(j), (kJllO

PHxy 4 m. P4/m(c), (k)OOl

Pi+xy (z) 4 1 P4 (d)

U""-. .P^I2xy 4 m. . P4/m(j)

51. .P^I'2xy (zj 4 1 P4^ (d)

4 ..P ^DI l>Qr|{z) 4 1 P4^ (a)

o CC C 1

(g) 4, . .P "^DI Ixy(z) 4 1 P4^ (a) 1 cc c

P4x2y 8 m. . P4/rrmn(p), (q)00\

Pt+x2y{z} 8 1 P4rm(g)

(^)100/010/001.

114 .. ) E

P4xx2z 8 . .m P4/mmm(v)

P4x2z 8 . m. t'i/mnmi s J , (r/ 22^/

.2.P Hxy 8 m. . P4/mcc(m) c

Q Vdnntr C- ( rl .2.P ^4xy (z) O 1 i 0 I H, /

0 Pd /vihm (m )C)-n . . 2C^xxz 0

. .mC+xy 8 m. . P4/mbm(i), (j)OOl

4xy [z) 8 1 P4bm ( d)

.b.C2xx2z 8 . . m t4/TnDTn { KJU2U

. .2I4xy 8 m. . P4/mnc(h)

. .21'4xy [z] 8 1 P4nc(a)

n . . mCM-xz a . m. r'i/ YlTWl ( i-y

. .2CIlz4xx 8 . .m P4/nrm(g)0\0

a m. .P Hxz U m Ilia P4 /mmc (0)00 1 Td J H c

. .2P I2x2y 8 m. P4^/minc(q) c

Q VA mr> ( -P ) . .2P I2x2yfzl O 1 ^ c 2

.2.P^2xx2y 8 m. P4 /mom(n)

.2.P^2xx2y iz] s 2 P4 om(e)

Q ITI m. . P M-xxz o P4 /mem (o)OOt c

Q riruit III/ . .2mxxz O ^/ \

.b2C 2xy 8 m. . P4^/rrbo(h) c -' .b2C 2xy{z} 8 1 t4 ^OOKQ) c

.n. I2xx2y 8 m. P4 ^/rmm('L)

.n.I2xx2y{z} 8 1 P4^nm(d)

115 . ) )

.n. I2xx2z 8 . . m P4^/mnm(j)

. . cIM-xz 8 .m. P4^/nmc(g)

4. . F2xxz 8 . .m P4^/ncm(i)lll

14- xz 8 .m. I4m2(i)

I4xxz 8 . .m I42m(i)

I4xy 8 m. . I4/m(h)

] 4xy(z) 8 1 14 (a)

1 V 1 (b) 4^ . . D2x]J {z] 8 14^

mx2y 16 m. . I4/rrmn(l)

x2y[z) 16 1 I4rm (e

I4xx2z 16 . . m I4/imw(m)

I'+x2z 16 . m. I4/mrrm(n)

. ,mC 4xy 16 m. I4/mam(k) c

. .mC 16 1 I4cm(d) c

.b.C 4xxz 16 . . m I4/mcm(l)0H c

. . 2 ^D4xz 16 . m. I4^/amd(h)

. .d^D2x2y (z) 16 1 I4^md(o)

7 .bdF 2xy (z) 1 0 14_L ^ aSOLi (b\ U J c ^

116 1

Tetragonal, tri variant (site symmetry 1)

The M--pointers:

PM-xyz P4 (h) P4xy2z P4/m(l)

The 8-pointers; m. . P 4xyz P4^/m(k)00l c P4xxz2y P42m(o) lC4xyz P4/n(g) ,2.P 4xyz P42c(n) c

n . . I4xyz P4/n(g)

, mC4xyz P42^m(f)

I4xyz 14 (g)

, cI4>Q?'z P42^a(e)

The" 16-pointers: P4xz2y P4m2(l) P4x2y2z P4/mmm(u)

, 2P 4xyz P4g2 (j) m. .P^4x2yz c P4/mac (n)OOl

, 2C4xyz P4b2(i) .mC4x2yz P4/nhm (n)

,2I4xyz P4n2(i) , cI4x2yz P4/nna(k)

Pi+x2yz P422(p) ,mC4xy2z P4/rribm(l)

>2^.C2xx2yz P42^2(g) .2I4xy2z P4/mna(i)

4^..P I lx2yz^ P4 ^2 2(d) 00 . .mC4xz2y P4/nrm(k) 3 cc c

(g) 4, . .? I lx2yz P4^22(d)OOl 1 cc c ^

. .c2C 4xyz P4/nca g) c (

4„. . I ^Dlxx2yz P4^2^2(b) 3 c i_ (g) 4^. .1^ Dlxx2yz P4^2^2(b) m. .P 4xz2y P4^/mmc(r)00l c

,2P I2x2yz^ P4^22(p) m. . P 4xxz2v P4^/mGm(p)00i c c

, 2^.I2xx2yz P4^2^2(g) , 22C 4xyz c P4^/nbc(k)

(^)100/010/001.

117 :

. .2I4xxz2y P4 ^/nnm(n)

Tn.2C 4-xyz P4^/mbG(i)00l c

.n. I2xx2y2z P4^/rmm(k)

. . cmxz2y P4^/nma(h)

. .in2C 4xyz P4^/ncm(j) c

I4xz2y I4m2 (j)

. .2C 4xyz I4o2(i) c

mxxz2y I42m(j)

.2.^D4xyz I42d(e)

mx2yz 1422 (k)

.2.^D2xx2yz I4^22(g)

mjQ?2z I4/m(i)

V a. . DHxyz I4^/a(f)

The 32-pointers I4x2y2z I4/nmn(o)

.b.C^4x2yz I4/mam(m)00l

.2."^D4xz2y I4^/amd(i)

.22F^4xyz I4^/aod(g)

118 . , :

Occurrence of Lattice Complexes

Table 29 Lattice Transfor- Multi- Site Occurrence complex mation from plicity symmetry space group, Wyckoff letter.

in all its standard (oriented) shift from standard representation represen~ representa- tations tion to cell

Orthorhombic, invariant

1 mrrm Pmmm(a) (h)\00, {o)00\, {d)\0\. 0 , (e) 0\0, {f)\\0, (g)0\\, (h)\\\

222 P222(a) , (h)\00, (o)0\0, (d)00\. (e) \\0, {f)\0\, {q)0\\, {h)\\\

P{z} 1 TWl2 Pmm2 (a) . (h)0\0, (o)iOO, (d)liO

p 200/010/001 2 . 2/m. Pmma( a. {h)0\0, (a)OO^j (d)OU a ,

P (z) 200/010/001 2 . . 2 Pma2(a) . (b)OW a P 100/010/002 2 222 Poom (e OOL (f)'20L (q)0\l, (h)\\l c

. . 2/m Pccm (a) (h)\\0, (o)0\0, {d)\00

P (z) 100/010/002 2 . . 2 Poo2(aj (h}0\0, {c)\00, (d)\\0 c ,

p 200/020/001 4 222 CmmcL ( a. 100, (b ) lOi ab

2/m. CmmcL (o J , (d)OOi lln f-p)lll . 2/m. Cmma (e j U J U 4 2

77 7T Ly A I / / 77 77 ^ . . 2/m Cxwrvmie > U * J ' 4 4 Z lln (-P)lll 1 Phcoi (e / Prnmioi lln (A)lll

P , (z} 200/020/001 4 ..2 Cmm2(o/ llo ab P 200/010/002 4 1 Paca (a/ (b)OlO ac , 100/020/002 4 1 Pbom(aj (b)lOO ^bc , Pbcf^> 100/020/002 4 ..2 Abm2(ay , (b)lOO 111 ^2 200/020/002 8 222 Fmrm(fj U U 14 2/m. Fmmm(Oy

.2/m. Fmmm(dy

. . 2/m Fmmm(e,

1 Cmom(d,

Cmoa ic,)

Coca (a,

119 Immm(k) Hi Ibam(e) HI Ibca(a), (b)lU 200/020/002 Frm2(h)U0

0 rrrnn Cmnm(a), (b)\00, (c)\0\, (d)00\ 222 Phan(a), (b)\00, (c)\0\, (d)00\

C222(a), (b)0\0, (c)\0l, (d)OOl

. . 2/m Pbcm(a), (b)00\, (c)OlO, (d)Oll Cfz] rm2 Cmm2(a), (b)0\0

. . 2 Pba2(a), (b)0\0 A[z} 001/100/010 rm2 Amm2(a), (b)\00 ..2 Pnc2(a), (b)\00

B 010/001/100 2/m. . Pmna(a), (b)\00, (c)\\0, (d)0\0

C 100/010/002 222 Caom(a)00l, (b)OU rbam(a)00l, (b)\Ol

2/m. . Cmcm(a), (b)0\0

. . 2/m Cccm(c), (d)0\0 Ibcm(c), (d)\00

1 Pbcn(a), (b)0\0

C (z] 100/010/002 ..2 Ccc2(a), (b)0\0 c Iba2(a), (b)OlO

002/100/010 . 2/m. Imma(c) Hij (d) Hi

1 Pnna(a), (b)00\

A [z) 002/100/010 ..2 Ama2(a) a Ima2 (a)

010/002/100 1 Pnma(a), (b)00\

2/m. . Imma(a), (b)00\

rrrnn Immm(a), (bJOll, (c)U0, (d) \0\ 222 Pnnn(a), (b)\00, (c)00\, (d)0\0

1222(a), (b)lOO, (c)00\, (d)0\0

. . 2/m Pnnm(a), (b)00\, (c)0\0, (d)0\\

I[z] mm2 Imm2(a), (b)0\0

..2 Pnn2(a), (b)OlO

120 E 4 mrm Fmmm(a), (h)00\ 222 Coca(a), (b)00\ F222(a), (b)00\, (a)lU, (d)lH

2/m. . Cmca(a), (h)00\

. . 2/m Ccom(e)llO, (f)llO

1 Pnnn(e)lU, (f)Ui Pccn(a), (b)OOh Pbaa(a), (b)OOl

F(z) 4 mm2 Fmm2(a) ..2 Coo2(c)U0

Aba2 {a)

0 8 222 Fddd(a), (b)OOl

DlzJ 8 . . 2 Fdd2 (a)

1 0 i raaaloj ^ (aj 222

Orthorhombic , univariant

P2x 2 2mm Pmmm(i), (j)OOi, (k)0\0, (DOW

2. . P222(i), (g)00\, (k)OlO, {l)0\\

P2x{z} 2 . m. Pmm2(e)^ (f)0\0 P2y 001/100/010 2 m2m Pmmn(m), (n)00\, (o)\00, (p)^Oh

. 2. P222(m), (n)00\, {o)\00, {p)\0\

P2y[z] 001/100/010 2 m. . Pmm2(g), (h)\00

P2z 010/001/100 2 mm2 Pmmm(q), (t)0\0, (s)\00, (t)\\0

. . 2 P222(q), M\00, (s)0\0, {t)\\0

(P2x) 100/010/002 4 2. . Pcom(i)00l, (3)011 c (P2y) 002/100/010 4 .2. Pmma(g), (h)OOi d (P2y) 001/100/020 4 .2. Pccm(k)00l, (l)\Ol c

(P2z) 010/001/200 4 . . 2 PcQm(m), (n)\lO, (o)0\0, (p)lOO c

200/020/001 8 2. . Cmma(h), (i)00\

002/200/010 8 .2. Cmma(3)lOO, (k) 10\

(P2-)ah 020/002/100 8 . . 2 Cmmm(m) HO Cmma(l)lOO

121 ... .

(P2x), 200/020/002 16 Fmmm(l) III

(P2y). 002/200/020 16 2. FmrmCk) HI

(P2z), 020/002/200 16 .2 Fmrrm(j ) HI

,2.P Biz 2 mm2 Prma(e)l00^ (fJllO a .2,P Blx 001/010/100 2 2. P222^(a), (b)0\0 c 2. .P Aly 010/001/100 2 .2. P222^(c)00l, (d)lOl c 2. .P^Alyfz] 010/001/100 2 m. Pmc2^(a), (b)lOO

..2P Cly[z] 100/001/010 2 m, . Pma2 (a) lOO 3. (.2.P Biz) 100/010/002 4 ..2 Pcca(d) 100, (e)l\0 a c (. .2P^Clx) 001/100/020 4 2. Fbom(c)OlO b c

(2. .P Aly) 020/001/100 . 2. Prrma(g)OlO c a .P (2. Aly) , 020/002/100 .2. Cmca(e) HI c ab

2^. ,CIlz mm2 Pmrm(a), (h)0\0 ..2 P2^2^2(a), (b)OlO

. .2^BIly 010/001/100 2 m. . Prm2^ (a)

(2. . .CIlz) 100/010/002 4 ..2 Paan(a)UOj, (d)HO 1 c

C2x 2rm Cmmm(g), (h)OOl

2. Phan(g), (h)OOl

C222(e), (f)OOl

C2x(z) . m. Crm2 (d)

C2y 010/100/001 m2m Cmrm(i), (o)00\

.2. Pban(i), (j)OOl

0222 (g), (h)00\

C2y(z) 010/100/001 4 m. . Crm2(e)

B2x 100/001/010 4 2. . Prma(e), (f)0\0

A2y(z) 001/100/010 4 m. . Amm2(d), (e)\00

(C2x) 100/010/002 2. . Cmcm (e) Coom(g)00l Ibam(f)00l (C2y) 010/100/002 Caom(h)00l Tbam(g)00l

(B2x)j 100/002/010 2. Irma (f) 111 (A2y). 002/100/010 .2. Imma (g) U 14 U

122 . 1

C2z rm2 CmrmCk), (l)0\0

..2 Pban (k), (l)0\0

Pbam(e), (f)0\0 0222 (i), (j)OhO

A2xlz] 001/100/010 . m. Arm2 (a) (C2z) 100/010/002 Ccom(i), (j)OlO Ibam(h), (i)0\0

2, . .C Fly m2m Cmcm(c)00l 1 c .2. Phcn(c)00l 0222 ^(b) 00

2,..C Fly(zl m. Cmc2^(a) 1 c

.2,.C Fix 010/100/001 2. . 0222 ^(a) 1 c

. .2.B, Fix 010/001/100 2. . 2nna(d) III ± D ..2,AlaFly{z) 001/010/100 m. . Ama2 (b)lOO

2. .P ^Flz mm2 CTma(g)OlO ah ..2 0222 (k)UO

.2P^^Fly 010/001/100 .2. Tcca(o)00l

,2P^ Flxlz] 001/010/100 .m. Abm2 (c)OlO DC

(2, .P , Flz) 100/010/002 . . 2 Coca (h) HO ab c Ibca(e)OlO

(.2.P, Fix) 002/100/010 2. . Ibca(c)00l DC a (..2P Fly), 010/002/100 .2. Iboa(d) loo ac b

I2x 2rm Irrmn(e), (f)0\0

2. . Pnnn(g), (h)00\

1222(e), (f)00\

I2x{zJ . m. Imm2 ( c) I2y 001/100/010 m2m Immm(g), (h)00\

.2. Pnnn(i), (j) lOO 1222(g), (h)\00

I2y{zl 001/100/010 m. . Irm2 (d) I2z 010/001/100 rm2 Imrmd), (j)lOO ..2 Pnnn(k), (l)0\0

'Pnnm(e), (f)0\0 1222 (i), (c)0\0

123

501-563 0 - 73 -9 ...

.2.B^A Iz 4 rm2 Irma(e)0l0 b a ..2 Pnna(c)lOl 12.2^2111Jc)OU

. . 2C Ix 001/100/010 4 2. . 12^2,2 Ja)lOl c b 111 2. .A C ly^ 010/001/100 4 .2. 12 ^(b) a c ^2 ^2 HO 2. .A C ly(zl 010/001/100 4 . 2. Ima2 (b lOO a c )

F2x 8 2rm Fmmm(g)

2.. Cmca ( d)

Ccaa (e) F222(e), (o)Hl F2x[z) 8 .m. Frm2 (d)

F2y 001/100/010 8 m2m Fmmm(h) .

. 2. Caca(f)

F222(f), (i)lll F2y[z} 001/100/010 m. Fnm2(c) F2z 010/001/100 rm2 Fmrm(i) ..2 Coom(k)UO

Ccaa (g)

F222(a) . (h)iii

D2x 16 2. Fddd(e)

D2y 001/100/010 16 .2. Fddd(f)

D2z 010/001/100 16 . . 2 Fddd(g)

Orthorhombic, bivariant

P2y2z 4 m. Prrmn(u),(v)\00

P2x2z 001/100/010 4 . m. Pmmm(w)j (x)0\0

P2x2y 010/001/100 4 . . m Vrnmiy), (z)00\

P2x2y(z) 010/001/100 4 1 I>rm2(i)

.P 2. 2xy 4 . , m Pcom(q) c 2. .P 2xy{zl 4 1 PGG2(e) c

124 ..

m. .P 2xz 4 .m. Prrma(i), (j)O^O a m. .P. 2xy{z) 100/001/010 4 1 Pma2 (d)

.2.P^Blz2y 4 m. Prma(k) lOO

2. .P^Aly2x(z} 010/001/100 4 1 Pmc2^(c)

.2.B2yz 4 m. Pmna(h)

2. .a:lxy{z] 010/001/100 4 1 Pnc2 (c)

b. .C2xy 4 . . m Pbam(g), (h)00\

h..c:Ixyiz] 4 1 Pba2(a)

4 . .m Pbcm(d)00l

.21P B C Flxylzl 010/100/001 4 1 Poa2^(a) ac a c

n. . I2xy 4 . .m PWLm(g)

n. .i:;^x/lzl 4 1 Pnn2(o)

2^. .CIlz2y 4 m. Pmrm(e)

.2^.CIlz2x 010/100/001 4 .m. Pmrm(f)

. .2^BIly2xIz) 010/001/100 4 1 Pim2^(b)

1.2,B, A FI Ixz 4 .171. Pnma(o)OlO 1 b a a 12,. C A FI Ixy (z) 100/001/010 4 1 Pna2^(a) 1 c a a

C2y2z 8 m. Cmmm(n)

C2 x2z 010/100/001 8 .m. Cmmm(o)

A5 x2yiz) 001/010/100 8 1 Arm2 (f)

C2x2y 8 . .m Cmmm(p)^ (q)00\

C2 x2y{zl 8 1 Crm2 (f)

.n.C 2yz 8 m. . Cmom(f) c

. .nA ^2xy(z) 001/010/100 8 1 Ama2(a)

2,..C Fly2x 8 . . m Cmcm(g)00l 1 c 2...C Fly2x{zl 8 1 Cmo2 (b) i c ^

125 ,2.F2yz m. . Cmca (f)

.2.F2xy[zl 001/010/100 1 Aba2(b)

n. .C^F2xy . m Ccom (t)

n. .C^F2xy{z) Coa2 (d)

.m.P ^2yz m. . Cmma (m) ab

m. .P ^2xz 010/100/001 . m. Cmma(n)OlO ab .m.P^ 2xy[z] 001/010/100 1 Ahm2 (d) be

I2y2z m. . Immm (Z)

I2x2z 001/100/010 . m. Immm(m)

I2x2y 010/001/100 . .m Irmrm(n)

I2x2y(z) 010/001/100 1 Imm2 (e)

b. .C 2xy , m Ibam(j) c b. .C 2xy[zl Iba2(o) c

,2.B^2yz m. Irma(h)

2. .A 2xz 010/100/001 . m. Irmad) a 2. .A 2xy(z] 010/001/100 Ima2 ( c) a

F2y2z 16 m. Fmmm(m)

F2x2z 001/100/010 16 , m. Frrmm(n)

F2x2y 010/001/100 16 , m Frrmn(o)

F2x2y[zl 010/001/100 16 Fmm2 (e)

d, . D2xy {zl 16 Fdd2(h)

126 Orthorhombic , tri variant (site symmetry 1)

The 4-pointers:

P2x2yz P222 (u) b2.C 2xyz Phcn (d) c

.2,P Blx2yz P222^(e) be. F2xyz Phoa(o) c

2^. .CIlz2xy P2^2^2(a) . maBj^2xyz Pnma ( d)

2,2,11 .FA abcabcB^C I LI Ixyz P2^2^2^(a) C2x2yz C222(l)

The 8-pointers: .2^,C^Flx2yz C222^(c)

P2x2y2z Prrmn( a)

I2x2yz 1222 (k) n. . I2x2yz Pnnn (m)

. .2C B^lx2yz I2^2^2^(d)lOl c b

. .mP^2x2yz Paom(r)00l The 15-pointers: b. .C2x2yz Pban (m) C2x2y2z Cmmm(r)

m. .P 2xz2y^ Prma(l) .n.C 2yz2x Cmcm(h) a c

2.2A 2xyz Pnna(e) .2.F2yz2x Cmca a ^ (g)

.2.B2yz2x Pmna (i) n. .C^2x2yz Ccom(m)00l

.22P 2xyz Paca(f) m. .P ^2x2y2 Crrma(o) lOO 3.C ab

b. .C2xy2z Pbam(i) c. .F2x2yz Coca ('I) c.2F2xyz Pccn(e) I2x2y2z Immm(o)

2.mP^ 2xyz Pbom(e) b. .C 2x2yz Ibam(k)00l be c

n. . I2xy2z Pnnm (h) 22.P22xyz Ibca(f)

j

2^. .CIlz2x2y Pmmn(g) . 2.B^2yz2x Imma(j) b (

F2x2yz b ZZ2 kJ

The 32-pointers: F2x2y2z Fmmm(p)

d. .D2x2yz Fddd(h)

128 , : . Table 30 Lattice complex Multi- Site Occurrence

in all its plicity symmetry space group, Wyckoff letter.

representations (oriented J shift from standard representation

Monoclinic invariant

0 1 2/m P2/m(a), (b)OlO, (a)00\, (d)\00. (e)HO, (f)0\\,(g)m,(h)m

P{xzl 1 m Pm (a),(b)OlO

P(y} 1 2 P2 (a), (h)00\, (a) iOO, (d) \0l

(^) 2 1 F2^/m(a), (h)\00, (o)00\, (d)\0\

(^P 2 1 F27o(a), (b)UO, (o)0\0, (d) \00 c

4 7 Li IIL i:^ J. Cj/ \ J 4UL-'_, \ J 1442 ^ ^ ^ab J

2 2/m C2/m(a), (h)0\0, (c)OOl, (d)0\\

C[xzl 2 m Cm (a)

Clyl 2 2 C2 (a), (b)00\

(^) A 2 1 P2^/c(a), (b)lOO, (c)00\, (d)lOl

4 1 C2/c(a), (b)0\0

^ e ^ ( ) F 4 1 C2/c(c)llO^ (d)U2

Monoclinic, univariant

P2y 2 2 P2/m(i), (j)iOO, (k)OOl, (l)lOl

P2y{xz] 2 1 Pm (c)

CP^Aly 2 2 P2/a(e)00l, (f)\Ol

CP^AlyCxz) 2 1 Po (a)

C2y 4 2 C2/m(g), (h)00\

C2y{xzl 4 1 Cm (b)

(^)100/020/001. (^)100/010/002. (^)200/020/001. (^)001/010/100. ( ® ) 100/010/102 From standard representation to cell.

129 IC Fly C2/a(e)00l c IC Fly[xz] Cc (a) c

Monoclinic, bivariant

P2xz 2 m P2/m(m), (n)OlO

P2xz(y] 2 1 P2 (e)

2,P^ACIlxz 2 m P2^/m(e)OlO 1 b 2,P^ACIlxz(y]lb 2 1 P2^ (a)

C2xz 4 m C2/m(i)

C2xz[y] 4 1 C2 (a)

Monoclinic, trivariant

P2xz2y 4 1 P2/m(o)

TiiP^2xyz 4 1 P2^/m(f)

2P 2xyz 4 1 P2/o(g) c

cA2xyz 4 1 P2^/c(e)

C2xz2y 8 1 C2/m(J)

2,C F2xyz 8 1 C2/c(f) 1 c

130 : Table 31 Lattice complex Multi- Site Occurrence

in all its plicity symmetry space group, Wyckoff letter,

qIh T "Fi~ "F"pnm c5i~3'nria"Prl 'PPr>'PP=5PTii~;^"t" "T on

Anorthic (triclinic) , invariant

0 1 1 PI ra;, rbjool, rc;oio, rdj^oo^ reJHo, (f)ioh (g)o\h (h)m

Pixyz] 1 1 PI (a)

Anorthic (triclinic), trivariant

P2xyz 2 1 PI (i)

131

1

00 o f<1 CN CN m (M CN CN (M I ^1 I O

rsi rH X N CD CO CN X H CTc X CM >1 B N X X •JD 00 X "J M ST N CM N CD CD Oj M 00 tN CN X X X N rH CN X CD N CN • N CNJ C-J N cO 04 M CN B cn X CN 00 r-i 3' CN B E N 0< CD CO X M N 1 X X • 1 X X X CM N E CO CN [14 CD CN cn N E CD >i CD xy CM CM CN 04 1 X CM H H X H -r~i 04 N •a CD £ m xy CN CD >. c X E CN CM CN 00 * X M iy\ * X N -H >4 CD N -r~i X rH CM H ^ CN cri N H 00 CO [14 CD N -H Tl 1 X \D "a' X CM CD < M X -rH CD 8(g 00 CD En cn X 1— X CM ra rH CN X CO X -H CD CN CN X X CNJ £ X X ^ CN X iH CO X .-H CN N * CO [l4 ^ CO CN In 'a- 'Si CN CN 04 s CD X N X- E rM CO CM CN >a- no CM CM m CM CN X CN csl N CH 10 cw x: "to CM N CD CD CD ro 00 VO Q M CD E ro 'J 04 CN cri X C4-I )96 X m X X N CD CO oo E X CN X IH >-13 H 0) X s CD X 4-1 00 m E X H X ro 01 M CO X ro CM >i X X * Ul CN CM ^ E * 0) X CM I*H N M >• CN X rM N CD C4-) CN E N U-) s CD N ro m [l4 CM CD Q CD CD I"* CN CO 04 0) CD M M 0) .H CN] tsl CO E CNH rs] CN X S r-H m X E e" CM CO N X CO X rM * X CD X 1^ B ^ CO Q) [n (U E CD Q 0) a 1^ M -a" CO CM CO r-) 5' CO r.) rsi 3 s CO '5

CN CN CN

s UIUI' rH E CM E- E E |CO |CO E [14 E -a |CO 1-0 in a B a i. o u \0 CM u CD CO S B B "o iro ft CO Eh rM 1^ ro CD (N iH K "u * CN CO >4 £ E E CD E Q fO H E fO £ m [14 E CD E £ 2 m B irn £ B ro CO rv] E Csj "ra M M 15 IS D 15 M fO M CD CD [14 Q 00 H rs) rH

-0 ro m E c B B a [14 [14 [14 a. 04 [n

133 CM

I I

EH

>i X H X X CN rH CM N CDM

(fl CN >. U a CN nH c X X CN £ e >1 N CO CN CD X X M X •H CN N H CN] 3 >, VD N CD X CN CD C?i CN (LJ e N c X CD o X CNJ CN Pj p N C o e c£) X X H u CM X CNJ ho CN N ri CD CO rH X T3 a- CN CD * !h N N mm CD CN] fc- CN

c*H

CN] N e mm CNl 2, rH >^ CM -P N X '5 m N CD CO CD £i N N H a' 3d 00 CD CO CN ro CN) rH 3 'S CNJ m e 00 'x X m e X N X X ID CN CN X X a- cn CN (1) X * Pu ro ro s X 0) s "J 2 CD CN rsi rH i-H

^1 1^ 1^ mB X H u CD CO

6 K 1^ CN 3 1^ CO CO 3

E .'^ It H Id IB 15 cn 3. CN (N CM

o

0< pC4

134 1 -

o00 o 5f CM D I o

X N M CN CM

rH 1 ^ X X

X -r—\ r-t N X in . 2 •=J> CN

1 X X H X CN >, c^ u X * CN X CU X H • CN + 0) CM N CN N o CM , fN ^ — M 2 Q £ m I n CN

CN CT) 1 X . X X X rn fN X * CN IX CN X i-H CX) X + ro X Cu CM •!T CN ct~> •tJ r-H H 0) CN CD CN CD Ix CN CN U5 X =f X m X — ro O X X 4-1 >< N X CN CX) + CN CN CX) CP s (N Cu CN > m N CN 2 CM fl^ ro CO CN CN c>n a •3' H S' CN N ~ CD a" X O. N X CU X cu Cu X CN s X m ^0 CN X CU a' X EH Q OJ X CD N QJ CN CN X CN H CN CN CN on CN CN c^ CN CN + In CN (N T) CN In S CN CN S E-i CO > CN 1 1 CN CN s 2 3 CN n CM rn CN u "o In 'u u "u E-i > ID + CN m CX) rH fH

CM CM Q CN a< m Cu m m ro 3' CN s 1 m H CN 'i5 CO * Cu Q >< rH CN CO +

CN CN CN CN CM cn rsj 1-4 a' (u M

135 .

o o IT) o O O CN -'^

J>1 .-H X rM N .

CM

(N N

>1

X •H CN a N •H X A! CN r-| N >. CN CN 0* ^ X N CO •H X -H 4J CO M • O (X CO (U X U C CM N X 'n H CN CN I X CP CN CN e CO ro "rr. X J X Mh N X m a COM X X • — X f-j X oo (1) ro tM CN X (N N -X i >! tN 1-2 X N 0) CD N M-) ro CN CN Q CO rH 3 N i u \D CO CN [l. a' CN X CN X m CN CO X N N 0) Q 10 2 X no CN CD to CM 2 CN OJ s O Icn* 2 E-> XI -a' |ro CM s "u X X X X ro CM CD ro on X X (N (N rs] In CN 3 O "o U Eh ft 1^ — CD O 00 CO tH XI CN CM

ro CN Q m Pm |ro* e fN Q< 3 3

'(0 ft. Cm "is Q Vn CO CO

n n n (0 0< HI a< H

136 in 00 01 01 01 Ol 01

I

Eh EH

>1 »< CN n N rsi rH

HOI

0) XI c

rH X CN o N 00

a Ho N >i

01

Eh in X 1 CM rci Ch ItH * rn CN rH CN m CN §

(0 X CO >i X N 03 X n>i O X X rH X X X X "a* M T3 .H .H u -Ic >< bCN neg CN

CN

CN •a EH N m>i X CO X X N rH -H > CN CN .H 3 c^ •H CN CN CN rH CN CN CN CN m CN X X 3 X .H K >H CO CN a. "cd

cx> cd CN

CN

CN CN CN 04

137 -

H

I

Q Q

X

a.

=> rH 0 X -H ID e -P Q. CN cu E •H QO U I X (0 X O

(TJ

1 X OJ X X CN m CN ^ (J — U CN u — " CO u CN 0 H — CO E * CN i-i S C/J CN c r; 0 X rlJ -H CN ro o 4-1 -rH CN O o O CN Cu E (U CN CN CW J-l E •H T3 N • 0 CN O e >i *• >i u o CD \ J-l w CN -P X o • i O N cu CN u >i >i T3 CN 1 X X CD CU • X rH £ >i e e m Vh a, CO M jj +J N U LO o CN 0. >i o w CN CM E -H u N CN (U CN O c >. a< — N >. O 0 CN a CN cu CN • X Cu X rH u fO CN m a, o N tu Cu i: w CN J >^ CN Si ifl n rH o u O IcD O N m N o ^ u O

n "u 1 X

OJ 1 X H X e' rH 0) X ro E CO CN M 1 1'^ Cu o O M 2 Q- N CN o U |cD Ccl rv4 rH o — X E E m u o a N u o c CN CN w ID 2(c) X "c >1 CN CD X CN Ch CN CD e' rH rH O o Cu o e CN CN CM o , e rH IcD Cu |cD

1 X X CN O o >1 1. CD O o CN Cu >^ CN 1 X o CN X o O U X n £ o m W B rs] U e" o E X g Cu cu •l—l CD CN •n CD 0. u CN CN CN 1 X Oj (a) X rH cn CD o CN O o 1 Cu CD o H CN O CN o

E o o E \ f.

CD CD CJ3 Cu 0< CU

138 ' ~.

CO o CO CO CO

I - J3 m CO CO a Q D

i mX O ft CN N N CM CN CN > f-{ rH >i O X 'o O CO c 1 X 0 X W iH m CM CJ -pt rH N ft -rH CN >i CN CN u n CM o N e u X CN 01 ft ro O CN ft , o CN X >i X ft g ro o CN • m o O • e Ix o ft g >^ o m CN o ft " ID g CN O >^ X r-i o ."^ ro ft ID 1 X u e X CM ro in ft in CN : ft •— N o CN •H CN (0 X '— "x c m -H ro o ft ft -H U2 ID 4J rO u +J C 'n CN dJ 1 X •H X -1—1 ft CO m N o ro o ft o CN IH N Cd tN e a' ft m O u X g CO -rH m CN a N ft g CN ft o '— o O ro ft N 1^ O CM CD ft U X in ft -i-j • ft ro .-IKl

e w CJ ro llD CN N o S Si CN CU o 1 — ft CN

o ft -— o o O o CN o

CN u o oft u CU CN O |iD ft — rwo —X> g O CN o ft ft o |in o u o 3

o "ca ft ft CN

CN 2 llO1 llD ft 0^ ft ft

139

501-563 O - 73 - 10 n ^ m vD GO CO 00 CO iH M .H r-t —i>I CN>I I J->I U3 U3 U3 O U U O

c Ho 4-1 a,

X

c c o •H Cr

<1) C " O Ul 1^ •H T3 P C O -H (u ,!>: Ix X •H (N N >, o

X 13 c o O CM H CM -H 4J X U s

e

01 4J

01 s — B X IS Ul c o 0 —^ o Id z — X B 01 01

Ul

10 u

o m

o ja 3

c to

0< 1fl O 10 o 10 04 — o. —

o O s o CO

Oi 04 0< 04

140 ,.' 1 ^'

CD O CO CD

I I si Q

N >. >1 >i CM — CN X X >^ vD cs CN CN ft iH Ot OJ

+ 1 U O 1 X d, CM P-( X rH * C • CM 0 rn rn •H o -P o a. 1 X 1 X -H ^ N o 1 X ot CM OJ >i U) X + 1 CN rH (D — X T! a- Ph CO •H •1—1 O r-t • CM CN m rO CM CN rH C X n CO c 0 ID CN •rH CP (i OC3 o X O a: rn 0) o —

1 X 1 X 1 4-1 X ><; X 0 c fN CN CO Oi w CN to' •1—1 X 1 o C u u CM 0 U) Of •H •a H CN -P c t £ U H Csl CO (1) M m m O ^ o CS CN* o o 0 N o o H-l H >1 0 2 — N CD tr >i ^ X X CN CN CO CN 1 f— CN p u rn CO CO in 4J CN N >i OJ z CN X X Ch cn CN CM CN W o ,— CM CDi iH O + 1 CO U) c 0 * _ Cm Cn u • cB CM • X E CM m CO N fO 2 — CN +J • X 1 X 1 X CM cu to X X N CM >, r-i rH CN 10 O u m • Oi OC I rn rH + 1 >X) CD CJ u ID N u w CN rH T3 rNj —a) o o JQ 0) u o U u CO to ^: CN CM Q. CM o E-i O N o XI r^J vD (N (N OJ CN S E CN (h CM o CN CO n CJ • '12 £5 o o .— o o CM ,— T3 o Oi CM ID* C s CM ~— 1 M ifl o CN O s Oo U — CN CM OJ o< CJ CO + m 1 CM o CN o tN X CN CN CD rH o u CM CM o CDi CN + Q o o o CJ "io u CN CO CN CD "(0 u (J "io rH u CM CM ro m CN

CM CN CN CN CN (N CN CN fN J- 00 VO CD CD CO 0< CM CM

141 • ,

CM J3

CJ

ts} CN >.

CN (X r-t

>i X U \D CN u E >i ID X Q) O o W

(J tsi fN to >i — -a C >1 — c 0 X X -H fN to X O m 0) O

M-l >1 0 c X • -H . m m Cs] g' c X H n-) O 0 in ! -H W O -o \i) U 4J c B O -H OJ u ' H CN m o m — 1-J iw N o CU >, 0 o p>-l o U)^ N rsj nme Lon

o -— s & cti m • ID B i-l \ H tN TJ O N e o , • 4J a. — rn Ul -p N o £ >i CD E CN IH I (/) W ^ 3' 1^ (a (n c N rs] 0 • CT u H. CN to X E • N , m 1^

to tl • a O o "oj XI (U o 1 O 10 x: 3 En 0 CD XI O e 0 o S -— w o a CJ c CN 10 • o o o 1^ \B lo a, oO o

'ia CM

ID 1^ 0- 0<

142 CO Oi o i-l CN \D vo r- r-- r- r~ H rH 1-1 iH M

I I I I I I 4 CM C*> J- in a> u) u> u> U U U u u u

4-) ft

a0)

73 c c^ o •rH CP (0 X ro (1)

M-l O- C

01 C " o m +J c O -H X — u H Cs] ft T3

>i o

+J 0) 0) c

10 U (U 'n "n r~- ^4 H >< >i T3 X rH X >i g CN (N X 1 Oi O OI rH +J U + 1 w in JJ u u o O >i 0) CN ft ft ft Ij .H CNl to in m m i ro Id 01 rH T3 i3 r« E-i 0 N CN "n £ Oi cDi o + 1 U tN w CN a c "n to >1 >i X X rH rH U U Oi Oi + 1 u rH o w W o "rO 0 u U ft ID ft

ID CN c>n rH "n CN CN in "cd ft CM o u u rH ft m ft ft CN

ID 10 kD U3 ft ft ft ft ft ft

143 -

r-l

I I CM TJ 01 ra J- Vm "TO a Q Q Q

>. (N to X

a, -H

S N X vD N >i a 1 X 0 X H ro r-t 4J U Q< U H

-H 1 X X • u (N 0 rH in a •1—1 •a 0 0Q 0

-0 0 a 01 - H n) 1 X — X X -H N i n x: IN X rH lU a a< CO 0 •H U CN m" , . CN e a (0 X c rH 0 •0 * 1 1 X -H c X +j H N 0 £ N u 0 « CO >. (U rH IN ^ CN X CO •0 i 2 CP X IH u m- cn X 4-1 c E X Cr> U IN C ID (1) 0 v. a< CN « CO •H CN CU ID IN 4-J , 0 0 0 • in 0 0 in 11 — U 1^ 0 r» r^ 0 "0 N cn CN E W M-l X 1^ X I4H ro jJ 0 Ul 0 — o IrH IK 00 a ^^] >. o rH (/) z Q c 0) Ul m I rJJ 00 c a' 0 rH cn In e IB B e CN a) S X • D^ 2 u CN 01 •p N m 'J n 0 X 01 i Ch n Ul w s u CN u S s s CN N CN '3' CO U rH g rH ro w u CN 0 ID S CN s Eh ja m IN 0 0 S u 0 m ro N CN 0 CN « c M e tti B CN 10 CN m N m rH N CN u CN CN ^0 CN u IN rM CO 0 ro n Q< e 0 15 IS 15 0. OS 0 0 1 G g 0 , o 1^ CN |cn CN U5 o ° J3

1^ |r^ 0 0 Id cu HI

CN ro

rH

1.^ 0;

144 o in

KD > cn U

-P

a0)

I X X

T! c Hc o nj X

o c

c 0 in N H -a p c i-H >| u >1 X i o m s U CU n "i ymmd m tior

0) e ec rH u ro •H g rH Q) >, E 'n' 4J »-l g rH m P N >. 0) "n "o

U2 > >. rH 1 X no X X X CO rH U g' m m ro m rH CO n O W S « C 1 o U U 0 o Oi g 10 on

(0 H >1 T3 X rH XI 0) m g* <0 43 N o s tH O m

N 1 x" CO X 3 rH o u 3 C CN o o « .-In w T! C rO

g

I—. m N (0 " rO " rO o 10 U •— a. CU i>l a. (N

O O H E ro n m a;

145 . - 1

o\ o iH m (N in ^ lO in irt in in m H H rH rH fH rH M

< I I I I I I * CM ir> J ID ro CO fn rn n D Q Q Q a Q Q

(1) O 1-1 3 O s

c 0 N H >. -P ,— O) rH u n u

0) -a a a

-H 1 X Id .— m c 0 cu ml c 0^ •H (0 y X o m 0)

1-1 c N rH

0 H 1 X >i X n CN Hh 01 ,— ro X o c Oj m CO ft 0 T) OS rH 4J H O , >i pVl m CN >i c mX 0 0 1 X 1 X M N 0^ X X >i 1 -p rH CN CN u CN X X in o + rH rH CD rH Oi cx -H 0. 1 rH + 3 X U u u O 00 CN E CM Q> ID P 4-1 o 14H CM o 0 m c^ >. N 0 0 0 Si • (IJ

I X c N X o m ^ rH H O u U 1 c i-H OJ CN a- O4 U tN N x: CN CN 0) 0 -p m mCM s c 0 0 0 UH XI 0 0 0 O +J a m N 0 o IS CN u •H rH 4-1 rH CN H O a o

CM tJi-w m n c o la H CD Oj o 0 4-1 rH r-H 0 o 0 4-1 >,

146 00 1- a"

I I I I I -H -H CM -H J- m ro m U U U U

N r-H X

ft

Z

O M-l N O >i 18(f X «

in o

—- -H T3 l3 N X e W n P VI •H in -P m >i 0) O 2 S u U3 § IN ITi 1 I

0) in a 0

Id ^ g (U V X 4J in >i • 05 N m m N

t< o

I 7a .H >i X 1^ ' Id o O X!

>1 >i X X rH rH cx "id 1 + u u ft m ft

Id Id (N — ft — OS ft 00

U3 ft

147^

501-563 O - 73 - 11 o

r-t

I I in I en CM -H 1- CO Q U UCO

C o •H -P H O

N X H X CM a. .-H x: o e o e N X X ID £

rH X! p10 I X X o c o•H (U 0 (U M Q) u <1) (1> XI 1 X •H X P N rH C >. >i H 1 X X X CM m 0) KD (U JJ C 0) IS S in •H -0 'J' M 0, „ 0 s o1 E o PC P rH w la c dj 0 •p tn p I X ~ Id 0) X v 0) rH u X CO X CO , O Ij P -P X rn N •H 4-1 cn S m X rH o e X Ij a, X en CN a X c-i X rn X H sr X — X 01 01 X "u CM X o X o H 4-) (N — CM 04 CN o x;

0) -H x' e CM X T3 S M m X 0) C H -H to tg 1.^ r-) 0) CN Oi in X> X! a 3 in O H in O CP ^ Ico 0) >< 0 Ifl 1? 10 H nS a. Q< U CN m -H

01 U 01 0} ll^ IcO x; x; E-i S (X. 0.

148 >

CN rH

I -5 J- jc in J- Q Q Q Q

X X CM X O u H X o — g X X CD "3' '3'

ft.

X e X CM X "a- X a e u E o rsj X! o N • g' IN U N 0 £ CM O H N g X >. (M o CO oj X — oCJ ~ X -H ID 04 o — X u X N N u rsi 04 X) u X e o o 04 0 o -p CO 0 rsiU O N N (N rs] .mm X 04 N 'J 'J IN rsi 04 CO u 00 0 m 0 rsj in J- N CN Q u P^ O g 1

N 'J U5 >i N 0) rsi rs| X 04 10 N U c •!T 0 >s o g U rsj ft. X N g "c <8 rsj o g" u 'c M U SD J-l g o rH ID (U sT O .mm Eh o U 0 g 0 0 o CM u X a, 04 1 X 04 00 0 -a U u g' J3 ft4 18 _g rs) CO u X g" u — 0 0 o u rsj rsi 04 g 04 0 15 0 rsi o o u rsi o 0

rsi 04

X rs] X g' U rsi 04 o u o rsi o 04 — o h-i rsi 0 o O XI •-lot 0 £ O XI O " s >4 o X "x X u O4 M g 04 u 0 CO ra 15 O 04 0 u — M 0 o o CM o

rsi rsi o g u <;r o XI c c \c HC a' 04 04 0

149 —1 1

1 to x; 1-^ 4: m j3 r-i Ah -< JS J- J- J- ^ J- -5 Q Q Q Q Q a Q

X X tsi X N CN X E >l (N H CN >i (J X CM 11 g o N CN u o M X 0 CN fN a- 0 CN u e rH

^ X CN CN X X (_) X CN -H M 0. U r— XI n -I E CN CN 0 GO 0 N u >i X 1— u X o CN CO (J 7? fN u u N X ID d, CN X CN (N N u • X u 4x e' rsj 0 'J r-t • 0 u CN 0. 0 E -H M 0 U 5 CO CO cu CO e" CN CO CN X o 0 X X 0 0 tN rs] CM (N N U U CN A o N X • N CO CO £ X Q E 0 0 X u 0 11. E 0 o * C3 e (J CN "o N u CN e • CN CO u • 0 CN CD CN N , >1 (N a. iH (N U N N •3' N CO X X N O CN X U u >l CD M Cu CN M 3' X fN E X e' e \ CN CO u CN 0 S3 E 0 CD 04 \ 0 CM CO 0. CD CD CN o CN X 0 O u tl) CN X O CN ^ 0 X CN 00 u CN H J X 0 e U CN 00 N .'^ -H 1< 0 2 M • Si fN U • 0 0. CN fN) 1^ 0 CO 0 0 CN U 0 E (J X O 0 X u fN CN o N X 0 0 CN tM f-t CN 0 CN U CN M H e' 15 0 u U 0 0 U CM o a. E 0 CN U CN 0 CN CN "x ST CN o a' CN 0 OJ cu 0 H« C^J u 0 0 u CN CD CO e 0 o 0 0 M e ST o (J o u o (_) 0 X o X X 0 CN fM 0 3 M 0 (i. 0 U CN fN fN fN CO 'I u 0 0 'S u u ci U s M (N 0 0 CO a' fN CD

fN E u e ll u XI c l-a' c 0 c c C c c \ c -\ c •1. -H H Csl H CM •H a' '3' a. 0 0 Qt< 0 Cn 0

150 —,

O CM 3' m m rH

I I I m CM -C £ w-t J- - 5 J- J- J- Q Q a Q Q Q Q Q N CN N >. >i CN X rH X X O -r-i CN u H e rH c

N >. N X X g X u Cvl •H ,— X u •H X • H CO X CO CM g e rH rU O U g' o r|a >1 o ^ (N N rH >1 E rs CN O X r-|(N •H N N rH X X 00 "J X o D >c rH >1 X g X X X g CN CN > CM X X CN "a* CO Pm CN g H CM u u U CO U CM rH u 3 U CM CO CN CM CN X) 00 ( o u CM o • X e N X e CN a ^ u CN 00 > 3 X H X X CM O X CP CM CM CN rM rH o CO Cu u CD tj OH u — CM u CD CO (J rH CN 3 O CH O CN rU e o CM \ CJ g X o ft CM CO X C4H H CO . C4H X N rH rH o CM CN CO rv] " >i 3 CM U CM CN CN CM CL, CN CN HH ft u X CN rU X CM CM X u ri:j CM CU CN CO ^ m M O X H rlJ Eh CN H cu > CH CD o H CD CM rH CD CO c CO (J N CM rH o "n H (1) rH M N g U CN 0) U CN X Cs] 3 U H O CO N "5 g* CN rH o N u O CN rH o CM CM CI) M 0) 3 'J' O D U , e CM CN O o > u CO S Si rH S CZD S U O g g N (J o Cn CN CN o u X o g X u Eh s "J > CM CM ci 5 M CD H rH o O (J S O CD U rH u 1^ o o 3 EH 3 CM > CM X — u CO u CM ft, rU CM o "cS o O o 3 H \ Q Csl u g > 00 , s o 1^ o CD O o CD 3 o 3 X! X "a-u 'J u 3 O a 3 In 3 H > a" CN

CM g 13 u o c 1 cfl c c CM •H o H O

151 CM 73 I Q

152 1 u —, ' -'

in\i)t*-coa^OrH CN .H

i H.H.HiHFHrHiHI I I I I r I Lo ^TJ r^TJ CO XJ "^"tJ o -•'O TU f-* csj Csl CM CM OJ .—• CN) .—•CM QQQQQQQC\J Q

N >i X rH u •n

CN CD

"n CN rH M N N u >-. X l—l N CN X f-i X X , , ^ — 'J' =r 3 • H 0 H M i-l ft CD — CJ o CN ^0 o • N N N g X >, >n X X H >i H N . ft a' -H CM o o CJ H -H N •n X -H "5 CN -H X 'J ""^ CO ^ "n M 00 CO • M H CNJ • ft u —CN N o X •r— £ X 0 ft X e N . CN (N u M CN X - U (N • • r ' X u c H u X CN CO £ N N 0 CO ft CN S 3 U-l ft a" o u o H '— CO o X X CN CN] X X H H X X X X X X X CM CN 3 X 0 ON N CN u H 0 CN 0 >i ft o CN l4_l 0 X ft X! c U Q 3 CN o -— N X s:- .-ICJ '— rH o X 0 N I4_j CN] rH H CD U O H CO 0 M CO tN N N O CN CN CN M CN X o CN 8 N X .-loi CN CN N X CD O M CJ CM CN rH u M (J ,— OD o N N 0 S CN CN 0 CO u H I'd-' CNH u U CN N 0) ft CN CN CN u CnI rs] — r-i CN u 0 s ft rs] CN u H 0 a' 1^ 2 0 CN CO o CJ CN o u 0 CNJ u H u o 0 CJ M 0 o CM O CM 3 0 a' a' 0 1^ u E ft u M u M o o I'd-' M \^ o ft o O 0 o o 0 L) o 3 -£ 0 £ 0 3

<0 10 'fO 0 3 U ft — u M M u M CN CN

CN CN e a ^ 1

ll 1^ 1^ 1^ I ^ ft ft ft CI4

153 o n r- 00 0^ o o O o o o o o o O r-<

I I I I I I -I > CM > > > CM > •-I J- i-l J- o a o u o u

in C

o

"J N 'J N CN CM N N N N N N N N N N 10 to 10 CO 10 (0 o "io Q "lO (0 O O M U M O, O M u > O CM a. CN CU CM CN Cvl u rsj CO

154 * — —-. —,

o CD CO 0^ 01

I — N I I I I I i J- J- J- Q Q Q Q

N f-l X 00 u — CM ' or-|(N 'J o oCM

»< tNH X O . •a' CM X N H o >1 o CN X cC X H H CNH O '— CM n CN CN CN O • u • ts] ,— 14-1 X >1 XI f~i o CN — N X CD >i N N N X , , CN 0) CN >. >! CN CN rsi o X H C X X tN CN M -— M X O X CN X X X O -H CO CN 'o^ H a- CN rH H X • Q •P PM u 0 O. O CN > CD O M M • M < iH 0) o r-l u u CN o CN u iH o O H CM CM 0 o - - 2 N • CO ro CN X CM CN CN CN X X M • r-\ O O N • « M CD O O £ >, c o CN CN M-l u N X tH .-ICN — CN m CN CN CN o > CM J- U M o X CN in Q O •r-t o o X ^ CN CO CM 00 CP • 1 O X X M CN M CN e X X (M 2 CO Q) CN N rH tH CM CN 4-> (N fNJ Q Q >: m CU > > CM 0) o >, o Cs] H • CJ U2 u N 1 X u u c CM CN X CN iH u (1) CM CM O CN CN It) CM N 4-1 Q c X3 CN X O > o m CN CO ^ IH O CN o CN (0 O O CM U O O CN N O rH Eh o H tsl CN CN CM O CJ CN X u CN N Q) O (N M Q) X in CN CN CD CN ro o U 2 N O CN Q 2 !h >i > 03 (U tN o X CN CN H CN H X O o CN u O H CN N o o CN rvj CN H 3 O u o u M O CM CN H CO 2 CN CN N CN CM CN ro N CN O CN U Q U CN M O M CN > O O U O o CN CO 2 u CN o CN o U CM o O CN CN CN CN X CM D CN u H CN M > CN U u CN X CN M CN CN o M CM CN CN O CN O CM o u O O o o u M o O CM (J O CM CN CN

3' O Q CM u O o M M > CN O o CN CN a"

CM CN CN CN ^ CN CM r-l (N CN

155 '

CO 00 CO CO 00 CO I I I r-1 s: CO si u-> ^ ID St o o

(N iH >i X

CO

CN >1 >i -H X , M o

.

c >1 >f X X — 3' H CO

N (N U Q O -H N N CN CN o >i u est X M O CO N 1^ CO (N C >1 Odt

IH

CN CO N N eg N fN H >. I X H i« 01 e -p D4: > lO tn _ 'I N H • o CN M \ N iH • C H • O o U CM CP CN • ta 0) CN !m N 4J a" CN 0) 00 o — O s o CN

bt l-i M > e 2 o 2 \ rH E o M U U u s C3 o "o "o > CO Q0^

M M Q 1^ O o > O o o O 3 o 3

rS U Q "(0 U — M H (N CN

1^ 1^ C Id c c Hc S

(1. O M o

156 —

U3 00 o CX3 I I I

a

o

c o •HP U 0) u

>1 N rHM U N >i « X

M CO

J 1^ N N OJ O H (N

c O o N CP nj >i ^^ X rH +j rN (D a) N M CN O 2 rsj H ft

X s 51' M O 2 'n' H CN CJ u u O r-IMo H O o

Tg N N u H >1 ft ft X o 3 O O O M o >° 0 o ft "to "to N ^ M N fd

> -3'

1^ 3' ft ft ft

157 1

CO

I I I

CM CNl Q Q

* >1 IN CN • 0< dt O

u N CM >1 >| CM CN 04 X < X CM o 04 CO N o E O >i u CM CM CO X rH >1 CN E CM e M 8(m X u CM a< CN Ch C x CM CN o" 04 fN ; o ou fo X >. (N CN CN a. P-i >4 10 CM c X O o o CM o 4J CL| fN O X u l Co X e CN CN o CN o u Ch N CN cr +> o X >i O •r-\ Cm fN CN tN O "~' X o M • CN X CN u CN (N Cu r~\ CN U 1—1 .-ICN '— -~- m N CU o ""'^ £ >. H ro o CM CN N U CM CO CQ 04 >1 H cd CN CM X -H M o X * CN cs N a. Q a. CP fN CN o >i I X o o CN CN e o 04 M CN CN O U O CM o O o X y-i (N (N N o U <— N CQ s:p CM m >, CM rsi o CM M 0 N 04 fd CN X o u Cm CM (N rH CM CN IM O 0! rH o 04 N CN o CN Oh o CN O 00 i« 04 e tu O O r-t M P e lO CO \ rsj E-i H Cm CD (d {J \ o CM oft N oCM CN u CM o N O >4 o 04 CN CM Cm N CN rH M CN o u CN m e CN (N ro O fN o o 11. o O ^ O CM s CM o O oa. O u o o< M o u o O O CN £t nS o o CM CM o CN CM H o s N o O o o X o o CM to E ft o >1 e •r-i CM CM 04 e e 'rd 'fO u 'to fN CO "to o O M CU CM u CM CN CN CN o

e c o o XI 04 04

158 -

I I I I I I I I I I o t-H CM m J- lox:QQQQQQQQQQCNJCN tDj3CNICslcnj3 CNlCM r-ijsCNJCNlrtx: ^-.-cCMCSJ^x:

CM O E CM N N N >i CN CN X >. >. rH >i CN rH X X CN U CN CN X (0 CJ h CN 0- C» CN CO CD fH rH CN M C o Si 00

>1 X B N E CJ >i X X * CN N ^' H X o r-l U rH CN CQ CN XI N o 10 o a. CU c rH E o N e M >! CD CJ '— M CN X CN * >, CN cu X CN CM O CN CJO (0 1 < X X U fN oo CN CN m CN O 2" H CM o O X 4-1 >i XI H O CN N 0 1—1 CJ H e XI O -— N << e • N CN rH u iH m CN M X) CN IB u CJ CN X o &i O |h N CU N n CN 2 CN >i 2 ta fN O CN H X tN a- 2 CN eg rH O O o O O U o WW (0 2 — CM CO XI N CN U \e OCQ H CN O e u S H 0! CN Cm "o o U rg o u O 2 IH o CN om O o OM O O HN O o H o U o "o CN u o OCN m 1^ cu |h M o o o O O o o 3 u 3 O 3 o N rH M u o 'cd nl as U X3 U (0 10 o M u — CN CN CM

IS E E c c c u o o c H o o a o o XI c XI X3 CM ft cu o CU

159 t —

in CO ID U3 I I I

x: f-f x; CN x; CN CM eg CM CM CN Q Q D Q Q D N >. CM X CM uU

c o o XI CO

>l X X s CN CN 0- CJ Cn UO CO c >. e >1 X CN CN CM CN U B CN X N t-l N CM •H >. O •1-1 CM Cn fN o o O -H u N O 00 "x 2 >. c CN a. CO X X o CN rsi •H XI 0 N CM CO 3 >. u CN Cu N X rs) u fH CM rH b CN >. N O N e XI iH rH O b E >i b< O o X (N -U X! CN • x: U CNJ X CO • ' U CN CN 04 CN s o U O (M CO o CN N O a' N O >i e X O fN O CN in >i XI e' o CO CO X e P. N X CN '>• 3 >i N CN s U CN X >. u U XI CN B CD N ^ rst rvi e o CO CM o CN XI o o 04 CN ca s U 14-) o CD (0 CO XI o 3 e c CO IB e XI nj CN < a. CN a. X N CM U X CN >, CN CO u XI CN u u 5" 10 g CM " CM CO 0< fc) (N o CO 10 E CO (N 01 o u E CD N 00 X r-H . •3 e lO e M M CN o CN 11. M X "o X u CN e XI CM 3 «: o u CO CO XI s 00 m CD CN X CM CM S b-, CD CN o >1 or N e o S o CN O CN o >, c CN o U E CM o CN XI CM Ir-i u CO CN CO o »— o o 3 o CU fN o CO O U CD XI CN o CD Q. CO

CO N CN o CN o o a. X3 e CO E CN u e o 0) rs] o o J CN fN o o CO o o o u o O o 3 HC1 o s XI CO

N CN CM CN u a. u 15 to CO O (J o 3 o u u CJ — 0. — in rg CD

CD ^ IS CO C o u -H E o u u O

160 t 1

ID

I I m GO CM CM XI CM M CM £ CM J3 OS CM Q Q Q Q

N CM Pj

rH N >i CN ,—1 X X CN CN CN r-H >i 0 O CM u >l •t— CM X CN XI vX) N H CP • Cu •H o £ o CO

CM X I>1 N CN X c X g H CO CN e u to o < e 1—1 — >. E • CO CM M CO B

00 N ><; >! CN >i CM M CO X r-{ X CM CM to CN (N D £ CN • Oj N r-i u >, ; N m N (N In CM g X CM CN CN o -H XI CN >1 >i 03 3 Cn CN CN o X H CO CN CO CD CM CN CO CM o U 2" 'n N fd CN CN N cn CO CO >i CM CM CN Q CM • ft CN Q \D < -H N (N CP IH CM H 1 o 01 CO CO CN O -r~i >^ CO CN CN O u CM ' • O CP X! e O CM CN IH CN CD CO X „CM CN Q hH CM 14-1 0< CP — o XI CO >i

H e • 00 CN • t>] O CN O CN f— X C O •— to • td CM < CM CN O 6h CM D X O OD CQ CM i CO e Q fN CN CJ Csl O CM rH O o Oa o CD o CN s M e CN CO 00 CN O >i E-i M-l tN o td fd ID o CN < r-i 2 (_) CN CM CN X o CM u g M XI s Csl a, CN CM CM (J CN U 'o u td o X u CO < 3 00 CN rH O -I-J CN o o o VD M CM o CN CM Q X! g' o o CN u Csl ft m o o 3 u o H 3 o __CM s 3 CN O CN

'

CM CM CN

C e a td T3 •H (0 0 •a O H

161 J 1 t11'

lO CM CN CN

1 1 1 1 1 1 1 I CM o > > > > > > ^ > CM CN CM CNl CM u U U U U U U

P*

P-( N E CQ X CN

* N f— >i P-j L) CN

P-I CN • o (N >1 1 1 CN N X

ni u H Q O IT b^ Xi •o 10 o ' eC 1< < fo CN u • CD X) cu * m , . CQ (N ~? iS X CM tN o N QJ iH Hi H U Kl * r— CN >i • w fo fo CN CLi CN * CN X fd

04 CN

ro V- TT c U • c CN Pj 2- e N fd

O >i CN Pj X >^ 0-1 CN 2 N CN CN u a' 'tt' U J3 Pj c ty ,-H U N X

Hi CM U "n fN N N Ti" '7"' N XI - >i X X d, 7^ * "77 D CN o — IN E >i 3' CU '~- • 1— (N O • M • CN £Q * CN «-i < • O CN N u • O • • o * i-|(M P-i 4-] Q fN N ' * ' CN 'T"' ft O >1 N • X o Si

'— O u 1— ' u ' td fd CQ «. o 1— u fd CQ n3 'n I--! Id CN N "io (0 ;* (N fi3 CN CN P-i CL| CN • CM H CN u CM

r—i (N CN CSJ (N CN CN CM O U td U G B o e o C E 0^ 04 P4 P4

162 1, . '''11 . .1'I,'11'>•1 —..',-'' —'1',p>1, ;,.' 1.'1 1 —', ,1'1 —.- 1'1. ' ., ' •1' — .•I1 '.' ,,>11'1 .'>

r~- CO O CN in m m •

1 1 1 1 1 1 1 1 1 1 CM J- CO cn o CM -« > ^> ^ > ,-.> - > - > rH > ^ > CN > Csl > ^ > CM CM CM CN CN CN CM CM CM U U U O U U U U u u U

1—

'—

N -— 1— CN M-l N >, — '— CN 00 N f-l U >. .— CN) XI — IB CD 1— ft X • '— B • >^ » .— CM • 1— X U -— X '— •-KM >— N rH '— o N f-t >i ,— o •— >. CM X Cs) XI - 1 • U aT CO N u CD • .— t-\ CN X Cm E c « •— XI "— fd 1— CN CM ; X u • — 1— rH N ^-^ fN >, XI • O CN ' 1— B O CD 1— X • 1— • X '— X CN '— N '— • N CN — rH U rH ns • 1— u. e < B u XI >— u • m ,— A X CM CN

1 ' — — ft .— — • u CN XI CN •a' rUO ,— o — O o O .— o o O N • 1— >i ,— (N CN 1—1

1 1 1 — > — — , X — X X X IB CN ID — 1— — 1— '— — rH "— o CN CM 0) • X — • ft • CN CO 1— „ CN M X X .— 1— td g XI u CQ '— CM (0 o tsl u E '— CN u • X .— E CN rH N as B U N X CN u iH e CM S • h-t CD ^ CO ! CN CM c X u CNJ CO X 10 CM <: CM o H u ja u 1— N ft s N '— 1— • CO 1— H • e >, X ,—

>— 1 — X rH CN — e N • CM .— IB

1 1 • ,— — . O < CO "— 1— CN < >i N e U ,— • o X r-H 1— CO • ' • X — o 1— CN , . >. u • • CM rsi CM — (—1 e CN >. o ' ' < 6. CO N •— .—1 X c CO '— ^-^ .— i-i • X U • u ,— Cu • CN ' u N O e f— 1-1 1— 00 — Si N • >, ft . 1— CN CN —u >. ,— CM < • X XI N <—i 'J' 1— ,— CN — O N N Q 1— >i 1— '— rH >— • i-H N iH • o >i >i ,— N * 1— O

1 ,— e -H • X XI — U CN U B • * • N CN rsl u

1 1 — CN — to £ CO ft , XI • < 3 O • X O XI N CM •— <^ -— o • CM 1— CN •-Ijf <— • • a- o ,— < N CN • CO 1-1 o r-|

1 ,— — p o N O< >. O rH • iu • CN CN CN CN u e •

, , , , o , , , , 1 , , , , • Id 'V. rd IM N N o 1d N 1b 1o 15 • .—1 1 — — — — — — 'a' U — — XI h b-, Q U — 04 u CN ft < CO M CN U a' <:

CN CN] CN CN CN rM rM fN (N CN tj 6 nj n3 (d u X3 X) XI e u u < < M M M

163

501-563 O - 73 - 12 '

CO rH

I I

N >i CN X fN u OD

X CN p4 CN O N N r-H CN >, X CN X CN X fa nH CN CO o to c o 0 N CN CN • N CN u fa o o M d) X N O >, V4 >i £ f—i CN •H _^ ^ N N u 0) rsi -— CN CN H Csl S X CN U fa CO CN CO CN M T 1^ X -H CO (N O CN n >i • CN CN CN N u fa CN CN rsj M On fN a< • O +j CN o o rvi >-|fM £ '— Q r;

u 1 >! M-l "xi fN N CN CN to

i o CM CN s O s 0. CQ O o a- N >i (X X U fa M

O o (J O CD H O >. XI CN M CU CO CD M cu o o r-i u fa M o o o O o XI O o m X o £ £ X N CO fH CN u fH < fa u >. CQ M fa o "cO rsi O o u a. CO Id CN CN "cO "io o CN CN CN fa M o CN CN CN -

CN CM

a.

164 .

o

I I CO J3 CM CM U3 X u U UCsJ

N X 0 CN a,

cs o&J £^ o

N X CM a-

a o a- >, p CM o Of N , >1 rs] u a. e o N N o X CN X u CM U 2 >i ft CM — CM Cn o >, X CM N N (J >, .H >i X -H X o >i CM CM 14-1 o fc. CH < CM >, •H -a- O O u CO ft IH CN u I CM >, g CM of u ft \ U o CM c X 0 s H >, rH u e CM <: Im ft ja O I- ft < N ft CM u CU CM >i 1^ 2 O ft CM X CD o CM CM o O < o O o ft o J3 ci nj ft (U o b £1 < ft ft ft o o 2 o o s o 2 O e CM ft o o XI 2 o ft o o .-1(N 2 o o o o U o o ft o 7) o o o "o o U < u ft ft — o u o o ft £ O J3 o o o o o o S o

ft J3 "(0 cd "rO u Id ft u CM <: o CM CM

o E \ o CM CM CM \ ft CN ft u ft CJ

165 I I I

o •H JJ O (U u •H T!

>^ U •P

CO o ^

u

I o c o s

EH

>i " £1 N >i — X CN " e U >i CM — O XI

>1 u <

O

o 0<

166 I I t-t -H o u

167 APPENDIX

Subgroup Relations

Tables 39-43 s — —

^3 Si

i H eg * ^3 00 (3 ^3 ^3 rC5 eg Ml CO CO eg Co CO eg CO '. ^ r-H CXI eg

CXI c\i CX]

eg H u PL, |ro H CXI H * --w — CXI ^3 '5 tv5 exi CX) * eg eg to CO r> CXI to Co eg ^ O CXI 'T3 CXI PL) t-s eg CX) PX, CO

I to ^ X eva CO eg r-H CX] \0O CXI CXI eg CXI CXI , CXI eo

CXI

05 r-H ~

CXI eg CXI rC) ex] CX] eg cs * eg exi i CX] CXI * 2 ^3 ea to CXI Ml eg CXI + Ml to X X eg c^a eg Ml eg CX) + 00 CXI CXI CXI Ml

CO

i eg ^3 lyj I-* * * ^3 ^3 CXI eg CO rCl * to eo Ml INi CO CO Ml eg 00 eg ^ eg v_ eg eg Ml H to H to H X to Ml X CX) Co eg eg Ml' 1^* I I Ml' Ml I CXI

^3 eg eg CXI H CX) CXI evj CO CX) fO eg t 1 * * Cj) * CO CX) CO 00 00 CO Ml Ml to to Ml Ml CO CX) eg IM^ I Mi

169 1^ i

fe4

&4 X)

CO

to H ^3 ^3

I] Cvd to to CM 00 CO CM o C\3 cva

cva

|N5 CO + to '~^ eg CO to CO CO c^o CO CO ^ c\ c0 CX! CXI Ml C^C1 CXI CO CXI

S3 Si CXI CO

CXI I H I H T CXI CO H CXI to to * : * to (31 CO Co 1^ * * CXI CO oo to Ml to Oo CXI Ol to Ml IMI Ml Ml 00 G IMi CO CX) Ml IMi IMi , ,

— ' I ^ CXI * CX) CO CXI to CXI CX) CXI S Ml CO to CO IMI S ^3 t>3 o, IMI :s: :^ S .-Ito to to i-la> .-IJ ,-lCM V_ CO •-IJ 00 to to CX) L.

S I to E s CX) CX) ^^ |to CX) Ml CX) IS: IS: ll to to MiTs .-|(0 --IJ CX) oo CX) CX) to I , '~~l J

170 I s s s

H

00

to

to Ml 00 X

— ^3 ""-i

to CO to ^ cva CO •-a X Po : X oa CM

^3 ^3 ^3 ^3 i S3 Si Si CM CM CM CM CM CM IH

I— H M to to to 'Ito to to X to * to * to to I I 00 * to rS? oo * CM * CM Oi * CM V_ * Ml H to CO CM to CM to CM X C\l Oi 05 PO « 1^ to 1^ 1"^ I'* CN) CM

I H H to

CM I ! to CM S r CM CM CM CM .-Im S CM S ^3 ^3 .-loo IMl Si' S3 CM CM CM H H •-13 — H H CM CM CM to CM CM -1^ -v-a CM CM H ^3 1 ^3 &3 S CM --W K! Mi CM CM CM CM to to to Ml to to CM N H CM H to to H CO CO CM CM 00 00 to to to CM Ml Ml Ml CM CM CO H CM -W- i-ltM CM CM Ml -13 CM to to

CM

^3 CM

^3 CM CM PO r I H < f H IH CM to CM CM to H CM CM*" CM E CM to CM CLi cm . H H H to H H n Ml Ml Ml Ml rid Ml Ml — CM CM ^3 ^3 ^3 P3 Ml to to to to CO 00 CO Ml CO H H Ml 1 Ml Ml 00 Ml to to to CM CM Ml •

171 172 to to

' to 35 S3 C-Cl

to S3 n to H to to to

to 03 03 H

CM i CM to

CM

to

CM

S3 CM

to

CM

CM

i33 CM to to to Q Ci to to 00 CM I CM 03 03 CM <33 CM CM 00 to

|to |to Eh Eh E-H Eh » CM Eh |v-h ,-ICM t-lCM t-lCM

,-ltM V_ .-lol v_ ,-|(M O

Eh Eh Eh Eh Eh to to 00 to r-H V-H

E CM CM to to C5 CM C) CM CM i-lM t-ICM .-IN -ICM O .-ICM 3 ,-HM o -|(M

00 00

to to r-H r-H Ml Ml

173

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

Bl ickrichtungen , 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, 40, 44, 47, 50 - form, 1 , 6 variant --, 16, 38, 42, 48, 51 - symmetry, 3 generating -- (G.L.C.), 16

Degradation Limiting site set, 4 cubic --, 9, 64

hexagonal --, 10, 66 Multiplicity

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

Dome (=dihedron), 7 Nodes, 1

17 5 s

Parallelohedron (=pinacoid) , 7 Space groups, 133-167

characteristic -- of a L.C., 11, 12 N-pointer, 3, 6 N-punkter Sphenoid (=dihedron), 7 N- punktner, 3

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

Symbol References, 26 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

9c p. 133, Im3m(0, ), second line of entry, in 6(c). 3m: replace 6 with 8

3 p. 138, P6„/mcm(D ), position 4(e)3.m is correctly given. [Note erratum 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 OiO with OOi

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 OiO.b.CZxx with 0i0.b.C2xx[z] p. 155, P422^2(D^ ), second line of entry: replace 4(c) with 4(c)2.. replace (d)2.. with 4(d)2..

4 p. 158, Pban(D2j^ ), under Pban add choice of origin : 0 in 222 p. 165, C2/c(C2^ ), 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, P42/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„/mcm: C 2. 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 (a) with (b)

p. 150, P42/mcm(D^^"'"^) : interchange (a) and (b); likewise (c) and (d) 20 p. 151, l4^/acd(D^^ ): interchange (a) and (b) p. 152, P42c(D2^^): 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-114A (1-71) U.S. DEPT. OF COMM. 1. 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 Space Groups and Lattice Complexes 1973 6. Performing Organization Code

7. AUTHOR(S) 8. 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 4Q2 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? Wfiat 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.

18. AVAILABILITY STATEMENT 19. SECURITY CLASS 21. NO. OF PAGES (THIS REPORT) Pn UNLIMITED. 184 UNCLASSIFIED

20. SECURITY CLASS 22. Price 1 1 FOR OFFICIAL DISTRIBUTION. DO NOT RELEASE TO NTIS. (THIS PAGE) $4.10 UNCLASSIFIED USCOMM-DC 66244-P7I

U. S. GOVERNMENT PRINTING OFFICE : 1973 O- 501-563